Recent zbMATH articles in MSC 11https://zbmath.org/atom/cc/112021-01-08T12:24:00+00:00WerkzeugAn equivalent form of the prime number theorem.https://zbmath.org/1449.110882021-01-08T12:24:00+00:00"Reddy, G. Sudhaamsh Mohan"https://zbmath.org/authors/?q=ai:mohan-reddy.g-sudhaamsh"Srinivas Rau, S."https://zbmath.org/authors/?q=ai:srinivas-rau.s"Uma, B."https://zbmath.org/authors/?q=ai:uma.bSummary: A simple proof is given that \(\sum_n\frac{ \mu(n)d(n)}{n}=0\) using the Prime Number Theorem. It is shown that this is equivalent to the estimate \(\sum_{n\leq x} \mu(n)d(n)=\circ(x)\) and to the Prime Number Theorem.Identities generated from the genus theory of real quadratic fields.https://zbmath.org/1449.050292021-01-08T12:24:00+00:00"Shen, Lichien"https://zbmath.org/authors/?q=ai:shen.lichienSummary: Applying the genus theory of Gauss on the real quadratic fields, we derive identities involving quadratic forms and genus characters.On the integer solutions of Diophantine equations \(x^2 + y^2 = p\) and \(x^2 + 2y^2 = p\).https://zbmath.org/1449.110552021-01-08T12:24:00+00:00"Tang, Jian'er"https://zbmath.org/authors/?q=ai:tang.jianer"He, Qixiang"https://zbmath.org/authors/?q=ai:he.qixiangSummary: In this paper, \(p\)-th order unity is used as the starting point, the integer solutions of equation \(x^2 + y^2 = p\) are given when \(p \equiv 1 \pmod 4\). On this basis, the integer solutions of equation \(x^2 + 2y^2 = p\) are given when \(p \equiv 1 \pmod 8\).Notes on explicit and inversion formulas for the Chebyshev polynomials of the first two kinds.https://zbmath.org/1449.110452021-01-08T12:24:00+00:00"Qi, Feng"https://zbmath.org/authors/?q=ai:qi.feng"Niu, Da-Wei"https://zbmath.org/authors/?q=ai:niu.dawei"Lim, Dongkyu"https://zbmath.org/authors/?q=ai:lim.dongkyuSummary: In the paper, starting from the Rodrigues formulas for the Chebyshev polynomials of the first and second kinds, by virtue of the Faà di Bruno formula, with the help of two identities for the Bell polynomials of the second kind, and making use of a new inversion theorem for combinatorial coefficients, the authors derive two nice explicit formulas and their corresponding inversion formulas for the Chebyshev polynomials of the first and second kinds.A fast algorithm for shortest addition chains.https://zbmath.org/1449.681482021-01-08T12:24:00+00:00"Wu, Jinxia"https://zbmath.org/authors/?q=ai:wu.jinxia"Wu, Chengxian"https://zbmath.org/authors/?q=ai:wu.chengxian"Wei, Kang"https://zbmath.org/authors/?q=ai:wei.kang"Liu, Bo"https://zbmath.org/authors/?q=ai:liu.bo.1|liu.bo.4|liu.bo.3|liu.bo.2"Li, Jinling"https://zbmath.org/authors/?q=ai:li.jinlingSummary: For the shortest addition chains considering integer \(n\), an improved fast algorithm is proposed, which uses the greedy algorithm idea to doubling from 1. When it is more than \(n\) after doubling, it traverses forward, so that the result is less than or equal to \(n\). Based on the Depth-First-Search algorithm, the current feasible solution and its depth \(d\) are obtained. When the depth exceeds \(d\), the current branch is no longer searched to reduce the space complexity, but the time complexity will increase exponentially when the additive chain is diffused. Combined with some pruning functions, the pruning operation is performed to reduce the time complexity, and then a better solution is obtained in an effective time. For the seven types of challenges, the improved algorithm is written by the Eclipse platform, and the number and its addition chain representation are given. The addition chain can be applied to modular exponentiation which is one of the core operations in public key cryptography, so the improved fast algorithm of shortest addition chain can improve the speed of execution of the public key cryptography.New classes of codes over \(R_{q,p,m}=\mathbb{Z}_{p^m}[u_1, u_2, \dots , u_q]/ \langle u_i^2=0,u_iu_j=u_ju_i\rangle\) and their applications.https://zbmath.org/1449.940762021-01-08T12:24:00+00:00"Chatouh, Karima"https://zbmath.org/authors/?q=ai:chatouh.karima"Guenda, Kenza"https://zbmath.org/authors/?q=ai:guenda.kenza"Gulliver, T. Aaron"https://zbmath.org/authors/?q=ai:gulliver.t-aaronSummary: In this paper, we consider the construction of new classes of linear codes over the ring \(R_{q,p,m}=\mathbb{Z}_{p^m}[u_1, u_2, \dots , u_q]/ \langle u_i^2=0,u_iu_j=u_ju_i\rangle\) for \(i\neq j\) and \(1 \leq i\), \(j \leq q\). The simplex and MacDonald codes of types \(\alpha\) and \(\beta\) are obtained over \(R_{q,p,m}\). We characterize some linear codes over \(\mathbb{Z}_{p^m}\) that are the torsion codes and Gray images of these simplex and MacDonald codes, and determine the minimal codes.A note on the hybrid power mean of the character sums and exponential sums.https://zbmath.org/1449.110822021-01-08T12:24:00+00:00"Wang, Xiao"https://zbmath.org/authors/?q=ai:wang.xiao.1|wang.xiaoSummary: The main purpose of this paper is using the properties of trigonometric sums and character sums to study the computational problem of one kind of hybrid power mean involving the two-term exponential sums and polynomial character sums, and give the precise computational formulae for it.A note on intersective polynomials in function fields.https://zbmath.org/1449.111092021-01-08T12:24:00+00:00"Qian, Kun"https://zbmath.org/authors/?q=ai:qian.kun"Liu, Baoqing"https://zbmath.org/authors/?q=ai:liu.baoqing"Li, Guoquan"https://zbmath.org/authors/?q=ai:li.guoquanSummary: Let \(\textbf{Z}\) denote the ring of rational integers. Let \(\mathfrak{L}\) be a field, and let \(p\) be its characteristic. Let \(f (x) = \sum\limits_{j=0}^n {a_j}{x^j} \in \mathfrak{L}[x]\) denote the polynomial ring over \(\mathfrak{L}\). Suppose that \(f (x) = a\prod\limits_{i = 1}^r (x - {\eta_i})^{e_i}\) over some algebraic closure of \(\mathfrak{L}\), where \(a \in \mathfrak{L}\), all the \({\eta_i}\) are distinct, and \(r, {e_1}, {e_2}, \cdots, {e_r}\) are positive integers with \(r \ge 2\) and \(n = \sum\limits_{j = 1}^r {e_j}\). The semidiscriminant \(\Delta (f)\) of \(f\) is defined by \(\Delta (f) = a^{2n-1}\prod\limits_{\substack{{1\le i, j \le r}\\{i \ne j}}} ({{\eta_i} - {\eta_j}})^{{e_i}{e_j}}\). It is proved that if \(n < p\), then there exist a positive integer \(m\) with \(m\mid{n!}\) and a polynomial \(G \in \textbf{Z}[{x_0}, {x_1},\cdots, {x_n}]\), which depend only on the vector \( ({e_1}, {e_2}, \cdots, {e_r})\), such that \(\Delta (f) = \frac{1}{m}G ({a_0}, {a_1}, \cdots, {a_n})\). This result is applied to investigate a question on intersective polynomials over the ring \(\mathfrak{L}[x]\), where \(\mathfrak{L}\) is a finite field.On the magnitude of the roots of some well-known enumerative polynomials.https://zbmath.org/1449.110462021-01-08T12:24:00+00:00"Rácz, G."https://zbmath.org/authors/?q=ai:racz.gabriellaThe Stirling numbers count the number of partitions of a set with \(n\) elements into \(k\) non-empty open subsets. Asking that blocks in the partition are ordered we get the Lah numbers and their generalization \(r\)-Lah numbers. The Whitney type variant is the \(r\)-Whitney-Lah numbers. With these numbers one can construct the enumerative polynomials called \(r\)-Dowling, \(r\)-Lah and \(r\)-Dowling-Lah polynomials. It is known that these polynomials have simple, real non-positive roots. The author gives bounds for the roots and compute their real magnitude.
Reviewer: Cristodor-Paul Ionescu (Bucureşti)Solvability of a ternary variable coefficient of Euler functional equation with constant terms and a perfect number.https://zbmath.org/1449.110092021-01-08T12:24:00+00:00"Shen, Jianghong"https://zbmath.org/authors/?q=ai:shen.jianghong"Gao, Li"https://zbmath.org/authors/?q=ai:gao.li"Zhang, Mingli"https://zbmath.org/authors/?q=ai:zhang.mingliSummary: Let \(\varphi (n)\) be an Euler function, this paper discusses the solvability of a ternary variable coefficient of Euler functional equation with constant terms and a perfect number \(\varphi (abc) = 3\varphi (a) + 4\varphi (b) + 5\varphi (c)-6\), whose coefficient is a special Pythagorean number by using the relevant content of elementary number theory, and proves that this equation has 39 groups of positive integer solutions.Constructions, counts and classifications of cyclotomic involutions over finite fields.https://zbmath.org/1449.111102021-01-08T12:24:00+00:00"Qu, Longjiang"https://zbmath.org/authors/?q=ai:qu.longjiang"Li, Kangquan"https://zbmath.org/authors/?q=ai:li.kangquanSummary: Since any polynomial \(f (x)\) over finite fields can be written uniquely as \({x^r}h ({x^s}) + f (0)\), based on this form, a new concept called the index of polynomials was presented in 2009. Since it has been proposed, this parameter has turned out to be very useful in studying value set size of polynomials, character sum, permutation polynomials among others. Involutions play very important role in the design of block ciphers. For the past two years, in order to provide more S-boxes for block ciphers, several scholars did some research about involutions. Recently, involutions of the form \({x^r}h ({x^s})\) over \({F_q}\) provided a necessary and sufficient condition for this polynomial to be involutional and presented a method to construct involutions with this form. However, the method needs to solve one equation system. In this paper, we firstly improve the method above, and obtain the explicit solutions of the equation system using the conjugate relation over symmetric group and the idea of block matrices. Secondly we give the number of involutions with any fixed index and constant term 0. Thirdly, according to the index, the known involutions with explicit expression are classified. Finally, we determine several classes of involutions, enriching the known results. Specifically, aiming at the involutions of low indexes, more specific involutional conditions of index 2 and 3 are given. For involutions with high indexes, using the compositional results obtained by us before, we give a class of involutions of the form \({x^r}h (x^{q-1})\) over \(F_{q^2}\).The singular integrals on the closed piecewise smooth manifolds of octonions.https://zbmath.org/1449.320082021-01-08T12:24:00+00:00"Gong, Dingdong"https://zbmath.org/authors/?q=ai:gong.dingdongSummary: The solid-angle coefficient method is used to study the principal value on the closed piecewise smooth manifolds in octonions, and a corresponding Sokhotski-Plemel formula is obtained. These results are proved to be useful in the further study of the singular integral theory in octonions.Rational points of algebraic variety defined by two polynomials.https://zbmath.org/1449.111072021-01-08T12:24:00+00:00"Gao, Wei"https://zbmath.org/authors/?q=ai:gao.wei|gao.wei.4|gao.wei.1|gao.wei.2|gao.wei.3"Huang, Hua"https://zbmath.org/authors/?q=ai:huang.hua"Cao, Wei"https://zbmath.org/authors/?q=ai:cao.weiSummary: Let \({\mathbb{F}_q}\) be the finite field of \(q\) elements. We study rational points on the algebraic variety \(W\) defined by two special polynomials over \({\mathbb{F}_q}\). When the greatest invariant factor of augmented degree matrix of \(W\) is coprime to \(q - 1\), we obtain the explicit formula for the number of \({\mathbb{F}_q}\)-rational points on the algebraic variety \(W\), which generalizes the known results.Counting solutions of a binary quadratic congruence equation.https://zbmath.org/1449.110042021-01-08T12:24:00+00:00"Duan, Ran"https://zbmath.org/authors/?q=ai:duan.ranSummary: Let \(n\) be a positive integer, \(Z_n\) denote the ring of residue classes mod \(n\), and \(Z_n^*\) denote the group of units in \(Z_n\), i.e. \(Z_n^* = \{s:1 \le s \le n \text{ and } \gcd(s, n) = 1\}\). The main purpose of this paper is using congruence conclusion and some results on exponential sums to study the number of elements of the set
\[ T(a, b, c, n) = \{(x,y) \in (Z_n^*)^2: ax^2 + by^2 + c \equiv 0\mod n\}\]
and give an exact computational formula for the number of elements of \(T(a, b, c, n)\).Reducibility of polynomials after a polynomial substitution.https://zbmath.org/1449.120012021-01-08T12:24:00+00:00"Drungilas, Paulius"https://zbmath.org/authors/?q=ai:drungilas.paulius"Dubickas, Arturas"https://zbmath.org/authors/?q=ai:dubickas.arturasLet \(K\) be a field, and let \(f \in K[x]\) be a polynomial of degree \(d \geq 3\) which is irreducible over \(K.\) \textit{M. Ulas} [J. Number Theory 202, 37--59 (2019; Zbl 1435.11068)] raised the problem of the existence of a polynomial of degree \(\leq d-1\) such that the composition polynomial \(f(g(x))\) is reducible in \(K.\) He proved the existence of such polynomial in case of \(d\leq 4.\) Here the authors solve the above problem.
There is an integer \(\ell\) in the range \(2 \leq \ell\leq d-1\) and a polynomial \(h\in K[x]\) of degree \(d\ell\) such that \(f(h(x))\) of degree \(d\ell\) is reducible in \( K[x].\) In particular, for any \(K\) and \(f\) as above, there is \(h\in K[x]\) of degree \(\ell= d-1\) such that \(f(h(x))\) of degree \(d(d-1)\) has an irreducible factor \(f^{*}(x) := x^df(x^{-1}) \in K[x]\) of degree \(d.\)
It is also shown that for any non-constant polynomial \(g \in K[x],\) the polynomial \(f(g(x))\) is irreducible over \( K\) if and only if for some root \(\alpha\) of \(f\) the polynomial \(g(x)-\alpha\) is irreducible over \(K(\alpha).\)
The authors also characterized all quartic polynomials \(f \in K[x],\) where \(K\) is a field of characteristic zero, for which \(f(g(x))\) remains irreducible over \(K\) under any quadratic substitution \(g \in K[x].\) This characterization is given in terms of K-rational points on an elliptic curve of genus 1.
As a corollary, they prove that the polynomial \(g(x)^4 + 1\) is irreducible over \(\mathbb{Q}\) for any quartic polynomial \( g \in \mathbb{Q}[x].\)
Reviewer: Piroska Lakatos (Debrecen)The solvability of the equation \(Z (n) = \varphi_e(\mathrm{SL}(n))\).https://zbmath.org/1449.110162021-01-08T12:24:00+00:00"Zhu, Jie"https://zbmath.org/authors/?q=ai:zhu.jie"Liao, Qunying"https://zbmath.org/authors/?q=ai:liao.qunyingSummary: By using elementary methods and properties for the pseudo-Smarandache functions, Smarandache LCM functions or generalized Euler functions, this paper studies the solvability of the equation \(Z (n) = \varphi_e(\mathrm{SL}(n))\) when \(e \in \{1, 2, 3, 4, 6\}\) or \(e\mid \varphi(\mathrm{SL}(n))\) (\(e >1)\). Furthermore, all positive integer solutions for these equations are obtained when they are solvable.The Diophantine equation \((x+1)^k+(x+2)^k+\ldots +(\ell x)^k=y^n\) revisited.https://zbmath.org/1449.110632021-01-08T12:24:00+00:00"Bartoli, Daniele"https://zbmath.org/authors/?q=ai:bartoli.daniele"Soydan, Gökhan"https://zbmath.org/authors/?q=ai:soydan.gokhanConsider the equation \((x + 1)^k + (x + 2)^k + \cdots + (\ell x)^k = y^n\), where \(k\) and \(\ell\) are fixed integers \(\geq 1\). In \textit{G. Soydan} [Publ. Math. 91, No. 3--4, 369--382 (2017; Zbl 1413.11074)] it is proved that in case where \(\ell\) is even this equation has only finitely many solutions where \(x, y \geq 1\), \(k \neq 1, 3\), \(n \geq 2\). It is also showed that it has infinitely many solutions with \(k = 1, 3\) and \(n \geq 2\). This paper, deals with the case where \(\ell\) is odd. It is proved that there is an effective constant \(C\) such that all solutions of this equation in integers \(x\), \( y\), \(n\) with \(x, y \geq 1\), \(n \geq 2\), \(k \neq 3\) satisfy \(\max \{x, y, n\} < C\).
Reviewer: Dimitros Poulakis (Thessaloniki)Goormaghtigh's equation: small parameters.https://zbmath.org/1449.110592021-01-08T12:24:00+00:00"Bennett, Michael A."https://zbmath.org/authors/?q=ai:bennett.michael-a"Garbuz, Ben"https://zbmath.org/authors/?q=ai:garbuz.ben"Marten, Adam"https://zbmath.org/authors/?q=ai:marten.adamThe title equation is
\[\frac{x^m-1}{x-1}=\frac{y^n-1}{y-1},\quad m>n>2,\; y>x\geq 2. \tag{*} \]
The following results are proved:
Theorem 3. If \((x,y,m,n)\) is a solution to (*) with \(2\leq x<y\leq 10^5\), then \((x,y,m,n)=(2,5,5,3)\) or \((2,90,13,3)\).
This theorem is a considerable generalization of two results by \textit{A. Makowski} and \textit{A. Schinzel} [Mathesis 68, 128--142 (1959; Zbl 0085.02902)], which the authors accomplish by introducing some new techniques and by taking advantage of both improvements in computational power and in the technical machinery underlying Runge's method.
Theorem 4. The only solutions to equation (*) with either \(m=n+1\) and \(3\leq n\leq 17\), or \(\gcd(m-1,n-1)>1\) and \(m\leq 50\), or \((m,n)\in\{(6,3),(8,3)\}\) are given by \((x,y,m,n)=(2,5,5,3)\) or \((2,90,13,3)\).
For the proof of Theorem 4 under the first condition, Runge's method turns out to be applicable (to an auxiliary equation) and, in fact, works particularly well, since in this case the Puisseux expansions are actually Laurent expansions with positive coefficients.
For the proof of Theorem 4 under the second condition the authors combine previous results by \textit{Yu. V. Nesterenko} and \textit{T. N. Shorey} [Acta Arith. 83, No. 4, 381--389 (1998; Zbl 0896.11010)], \textit{M. A. Bennett}, \textit{A. Gherga} and \textit{D. Kreso} [An old and new approach to Goormaghtigh's equation, submitted], \textit{F. Beukers} and \textit{Sz. Tengely} [An implementation of Runge's method for Diophantine equations, 2005, \url{arXiv:math/0512418v1}] and a precise implementation of Runge's method based on an algorithm for explicitly solving equations of the shape \(P(X,Y) = 0\) with \(P(X,Y)\in\mathbb{Z}[X,Y]\) satisfying ``Runge's Condition''.
The proof of Theorem 4 under the third condition consists in the explicit computation of all rational points on two hyperelliptic curves of respective genus 2 and 3 whose Jacobians have rank one. The solution for the genus 2 curve is accomplished by Chabauty-type arguments by using certain routines of Magma which are based on work of \textit{M. Stoll} [Acta Arith. 98, No. 3, 245--277 (2001; Zbl 0972.11058)]. For the solution in the case of the genus 3 curve the authors confine themselves to saying that they use Chabauty-type arguments based on work of \textit{J. S. Balakrishnan} et al. [Chabauty-Coleman experiments for genus 3 hyperelliptic curves, In: Research Directions in Number Theory, Assoc. Women in Math. Ser. 19, 67--90 (2019; Zbl 1436.11149)].
Theorem 6. There are no solutions to equation (*) with \((m-1)/(n-1)=3\).
In Theorem 6 we have the equation \((x^{3n-2}-1)/(x-1)=(y^n-1)/(y-1)\). The authors expand \(y\) as Laurent series in \(x\), say \(y=x^3+a_2(n)x^2 + a_1(n)x + a_0(n) + E(x,n)\), where \(a_0,a_1,a_2\) are explicit and \(E(x,n)\) is an infinite series in \(1/x\) satisfying \(0<E(x,n)<1/(3(n-1)x)\). From these facts the authors conclude that \(\Vert a_2(n)x^2+a_1(n)x+a_0(n)\Vert < 1/(3(n-1)x)\), where \(\Vert\cdot\Vert\) means distance from the nearest integer. On the other hand, by Theorem 7 of the paper (which comes from a combination of previous results of Nesterenko-Shorey and Bennett-Gherga-Kreso) \(x\leq 3n(n-1)\). From these facts the authors conclude that it suffices to check the inequality involving \(\parallel\cdot\parallel\) for each \(n\) with \(18\leq n\leq 3348\) and each \(x\) with \(47\leq x\leq 3n(n-1)\).
In the proofs of the three theorems above machine computations play an essential role.
Besides the interest of its results, the paper is also very useful as it exposes a variety of results and techniques that might be useful in solving various types of Diophantine equations.
Reviewer: Nikos Tzanakis (Iraklion)Open conjectures on congruences.https://zbmath.org/1449.110012021-01-08T12:24:00+00:00"Sun, Zhiwei"https://zbmath.org/authors/?q=ai:sun.zhiwei|sun.zhi-wei.1|sun.zhi-weiSummary: We collect here 100 open conjectures on congruences made by the author, some of which have never been published. This is a new edition of the author's preprint [\url{arXiv:0911.5665}] with those confirmed conjectures removed and some new conjectures added. Many congruences here are related to representations of primes by binary quadratic forms or series for powers of \(\pi\); for example, we mention two new conjectural identities
\[\sum_{n=0}^\infty {\frac{12n+1}{100^n}} \binom{2n}{n} \sum_{k=0}^n \binom{2k}{k}^2 \binom{2(n-k)}{n-k}\left (\frac{9}{4}\right)^{n-k} = \frac{75}{4\pi}\]
and
\[\sum_{k=1}^\infty \frac{3H_{k-1}^2 + 4H_{k-1}/k}{k^2\binom{2k}{k}} = \frac{\pi^4}{360} { with }H_{K-1}: = \sum_{0 < j \le k-1} \frac{1}{j}\]
and include related congruences. We hope that this paper will interest number theorists and stimulate further research.Correction to: \(X\)-coordinates of Pell equations as sums of two tribonacci numbers.https://zbmath.org/1449.110512021-01-08T12:24:00+00:00"Bravo, Eric F."https://zbmath.org/authors/?q=ai:bravo.eric-f"Gómez Ruiz, Carlos Alexis"https://zbmath.org/authors/?q=ai:gomez-ruiz.carlos-alexis"Luca, Florian"https://zbmath.org/authors/?q=ai:luca.florianSummary: In this work, we correct an oversight from [Period. Math. Hung. 79, No. 2, 157--167 (2019; Zbl 1424.11036)].A Diophantine equation with the harmonic mean.https://zbmath.org/1449.110702021-01-08T12:24:00+00:00"Zhang, Yong"https://zbmath.org/authors/?q=ai:zhang.yong.4"Chen, Deyi"https://zbmath.org/authors/?q=ai:chen.deyiFix a polynomial \(f(x) \in \mathbb{Q}[x]\). The authors consider the Diophantine equation for values \((x,y,z)\) defined by the harmonic mean of \(f(x), f(y)\) equalling \(f(z)\), that is, to the equation \(2 f(x) \, f(y) = f(z) \, (\,f(x) + f(y)\,)\). For \(f(x) = x^2 + bx + c\) with \(b, c\) integral and \(f\) without multiple roots, they show that should \((x, 2z + b, z)\) be an integer solution, then there are infinitely many integral solutions. Similarly, for a specific class of monic cubic \(f\) they give one solution for each \(f\). The techniques are Pell's equation, elementary theory of elliptic curves, and clever algebraic manipulations.
Reviewer: Thomas A. Schmidt (Corvallis)Note on a paper by Bordellès, Dai, Heyman, Pan and Shparlinski.https://zbmath.org/1449.110112021-01-08T12:24:00+00:00"Wu, J."https://zbmath.org/authors/?q=ai:wu.jianyu|wu.jiangqi|wu.jinhui|wu.jinguang|wu.jiadong|wu.junshuang|wu.junqing|wu.jundong|wu.junyan|wu.jinsong.1|wu.jianlong|wu.junchao|wu.jizhou|wu.jiahong|wu.jong-wuu|wu.jiahao|wu.jiajun|wu.jue|wu.jiansheng|wu.jinnwen|wu.jia|wu.jiesheng|wu.jincao|wu.jichang|wu.jiangling|wu.jian-hong|wu.jialong|wu.jianmou|wu.jiankun|wu.jiachao|wu.jianjia|wu.jianghao|wu.jilin|wu.jinming|wu.jongshyong|wu.jenchau|wu.jingshu|wu.jingkai|wu.junting|wu.jiekang|wu.jingyu|wu.jianmin|wu.jiatao|wu.jingna|wu.jingxiang|wu.jiaying|wu.jianxun|wu.jiaping|wu.juying|wu.jingtong|wu.jianglong|wu.jongyu|wu.jilong|wu.junqiang|wu.jier|wu.jing.1|wu.jesse|wu.jiangwen|wu.jinghe|wu.jiao|wu.jingli|wu.jinjie|wu.jongwuu|wu.jiaru|wu.jiming|wu.jichun|wu.jiangmin|wu.jie|wu.jianchao|wu.junjiang|wu.jiangqiao|wu.jiangwei|wu.jianzhao|wu.jianguang|wu.junliang|wu.jinqi|wu.jianzhuan|wu.jingshown|wu.jin-yong|wu.jiating|wu.jayne|wu.junhao|wu.jianle|wu.jianrong|wu.jingya|wu.jiongyang|wu.jinbo|wu.jianzhong|wu.junsheng|wu.junli|wu.jiabang|wu.jiangniu|wu.jiye|wu.jiabi|wu.jiaxian|wu.jianhai|wu.junqiao|wu.jimmy|wu.jiying|wu.jieyun|wu.jinlin|wu.jiangping|wu.jiayi|wu.jun|wu.jike|wu.jieer|wu.jin|wu.jianghang|wu.jiwen|wu.jieting|wu.jiasong|wu.jinwen|wu.jianghong|wu.jinlong|wu.jianhong|wu.jiabin|wu.jingjing|wu.jingwei|wu.jingxian|wu.jianhui|wu.jang-mei|wu.jiangzhong|wu.jiankang|wu.jiangzeng|wu.jiongyu|wu.junhui|wu.jianshe|wu.jibing|wu.jinqiu|wu.jihao|wu.jiaji|wu.jigang|wu.jianping|wu.jiawen|wu.jiangqin|wu.jitao|wu.jiawei|wu.jiangxing|wu.jiaojiao|wu.jiuhui|wu.jingkun|wu.jiarong|wu.junlin|wu.jiayang|wu.jiang|wu.ji|wu.junying|wu.jishan|wu.jianhong.1|wu.jingyan|wu.jieqiong|wu.ju|wu.jiezhi|wu.jintao|wu.jilian|wu.jinmu|wu.jiande|wu.jianan|wu.jianglun|wu.jiangfeng|wu.jingyuan|wu.jihua|wu.jinbiao|wu.jijiang|wu.jianfeng|wu.junjie|wu.jiayu|wu.jiajing|wu.jingbo|wu.jiachun|wu.jiagao|wu.jianguo|wu.jinyuan|wu.jinqiao|wu.jianbing|wu.jiening|wu.jingchen|wu.jinjin|wu.jinying|wu.jiahui|wu.juanjuan|wu.jiewen|wu.junwei|wu.jianchun|wu.jie.3|wu.jinliang|wu.jiachen|wu.jiyu|wu.jianbo|wu.jianying|wu.jieyi|wu.jian|wu.jingyi|wu.jigui|wu.jinshan|wu.jianxin|wu.jierun|wu.jinxia|wu.jingning|wu.jianshong|wu.jingbing|wu.jinghua|wu.jiongfeng|wu.jonathan|wu.jiancheng|wu.jianzhang|wu.jisong|wu.jyh-yang|wu.jingyang|wu.jun.1|wu.jingjie|wu.jifong|wu.junbao|wu.jinnan|wu.juan|wu.jinzhong|wu.jinsui|wu.jiahuan|wu.jiafeng|wu.jianqiu|wu.jianhuang|wu.jinjian|wu.jie.4|wu.jingwen|wu.jiongqi|wu.jianqiang|wu.junhua|wu.julin|wu.jiangmei|wu.jun.2|wu.janet|wu.jianjun|wu.jixiang|wu.juhao|wu.jingtang|wu.junwen|wu.jingpeng|wu.jiangang|wu.junfang|wu.jintang|wu.jie.2|wu.jianghua|wu.juanyong|wu.jianming|wu.jiaxiang|wu.jinhua|wu.junbin|wu.jane|wu.jijun|wu.jiechao|wu.jiacheng|wu.jiaochao|wu.jinping|wu.jingzheng|wu.jihui|wu.junxin|wu.jiaxi|wu.jianing|wu.jingzhu|wu.jibo|wu.jinzhao|wu.jinjun|wu.jingxin|wu.jianzu|wu.jianyong|wu.jiaoyu|wu.jieming|wu.jingang|wu.jianwu|wu.jingling|wu.jianshi|wu.jiayong|wu.jie.5|wu.jinglai|wu.jian-liang|wu.junfeng|wu.junqi|wu.junde|wu.ji-min|wu.junjian|wu.jiqing|wu.jiangning|wu.jiaqian|wu.jing|wu.jingxia|wu.jinpei|wu.jie.6|wu.jiandong|wu.jiantian|wu.jinrong|wu.jie.1|wu.jiaqi|wu.junmin|wu.jianbao|wu.julong|wu.jianwei|wu.jiancun|wu.junru|wu.jingcao|wu.jiajie|wu.jieheng|wu.jianbin|wu.jiangtao|wu.jiagui|wu.jianhua.1\textit{O. Bordellès} et al. [J. Number Theory 202, 278--297 (2019; Zbl 07063099)] present, among others, asymptotic formulas for the sums \(\sum_{n\le x} f(\lfloor x/n\rfloor)\). The present author improves some of their results sharpening the error terms.
Reviewer: Pentti Haukkanen (Tampere)Some notes on the multiplicative order of \(\alpha+\alpha^{-1}\) in finite fields of characteristic two.https://zbmath.org/1449.111132021-01-08T12:24:00+00:00"Ugolini, Simone"https://zbmath.org/authors/?q=ai:ugolini.simoneThe author proves some results on the possible order of \(\alpha+\alpha^{-1}\) for \(\alpha\) a non-zero element of a finite field of even order. The results are based on his earlier paper, [Contemp. Math. 579, 187--204 (2012; Zbl 1302.37074)].
Reviewer: Arne Winterhof (Linz)An application of Baker's method to the Jeśmanowicz' conjecture on primitive Pythagorean triples.https://zbmath.org/1449.110662021-01-08T12:24:00+00:00"Le, Maohua"https://zbmath.org/authors/?q=ai:le.maohua"Soydan, Gökhan"https://zbmath.org/authors/?q=ai:soydan.gokhanIn 1956 Jeśmanowicz conjectured that the equation \[(m^2 - n^2)^x + (2mn)^y = (m^2 + n^2)^z,\] where \(m,n \in {\mathbb{N}}\) are coprime, has the only solution in positive integers \(x,y,z\), namely, \((x,y,z)=(2,2,2)\).
This problem is still unsolved, although there are many partial cases that have been settled by now. In this paper the authors show that that Jeśmanowicz' conjecture is true if \(mn \equiv 2 \pmod 4\) and \(m>30.8n\). The proof involves an effective lower bound for a linear form in two logarithms.
Reviewer: Artūras Dubickas (Vilnius)Many solutions to the \(S\)-unit equation \(a+1=c\).https://zbmath.org/1449.110612021-01-08T12:24:00+00:00"Ha, J."https://zbmath.org/authors/?q=ai:ha.joseph|ha.junhong|ha.jongsung|ha.jaegeun|ha.jincai|ha.jaewook|ha.jinkyung|ha.jihyun|ha.jeongcheol|ha.junsoo|ha.jonghyun|ha.jinting|ha.jeongseok|ha.jingwan|ha.jaecheol"Soundararajan, K."https://zbmath.org/authors/?q=ai:soundararajan.kannanConnecting to and improving the corresponding results from the literature, the author shows that for all \(s\), there exists a set \(S\) of \(s\) primes such that the equation \(a+1=c\) has \(\gg \exp(s^{1/4}/\log s)\) solutions where all prime factors of \(ac\) lie in \(S\). In the proof, among other things, tools from combinatorial and analytic number theory are combined.
Reviewer: Lajos Hajdu (Debrecen)Yet another generalization of Sylvester's theorem and its application.https://zbmath.org/1449.110932021-01-08T12:24:00+00:00"Laishram, Shanta"https://zbmath.org/authors/?q=ai:laishram.shanta"Ngairangbam, Sudhir Singh"https://zbmath.org/authors/?q=ai:ngairangbam.sudhir-singh"Singh, Maibam Ranjit"https://zbmath.org/authors/?q=ai:singh.maibam-ranjitSummary: In this paper, we consider Sylvester's theorem on the largest prime divisor of a product of consecutive terms of an arithmetic progression, and prove another generalization of this theorem. As an application of this generalization, we provide an explicit method to find perfect powers in a product of terms of binary recurrence sequences and associated Lucas sequences whose indices come from consecutive terms of an arithmetic progression. In particular, we prove explicit results for Fibonacci, Jacobsthal, Mersenne and associated Lucas sequences.Some combinatorial properties of restricted Eulerian polynomials.https://zbmath.org/1449.110412021-01-08T12:24:00+00:00"Zhang, Xutong"https://zbmath.org/authors/?q=ai:zhang.xutong"Zhang, Biao"https://zbmath.org/authors/?q=ai:zhang.biaoSummary: A new class of restricted Eulerian polynomials is defined by restricting both the first and last letters of the permutations. The relationships of the new restricted Eulerian polynomials and the original ones are obtained in two different ways by using the property of the new restricted Eulerian polynomials. In addition, the equidistribution of excedances and descents for permutations with the last letter restricted is proved.Hyperelliptic curves and arithmetic functions.https://zbmath.org/1449.110742021-01-08T12:24:00+00:00"Davis, Simon"https://zbmath.org/authors/?q=ai:davis.simon-brianSummary: The precise form of the correspondence between Dirichlet series, modular forms and Riemann surfaces is given. An upper bound for the prime \(p\) following from maximum value of the number of points of an unramified algebraic curve of finite genus over \(\mathbb{F}_q\) is derived. The rational characters of absolute invariants of arithmetic subgroups of the modular group are verified and their role in rational conformal field theories is elucidated. The transcendental limit at infinite genus is examined.Arithmetic summable sequence space over non-Newtonian field.https://zbmath.org/1449.260022021-01-08T12:24:00+00:00"Yaying, Taja"https://zbmath.org/authors/?q=ai:yaying.taja"Hazarika, Bipan"https://zbmath.org/authors/?q=ai:hazarika.bipanSummary: In this article, we introduce the sequence spaces \(AS (G)\) and \(AC (G)\) of arithmetic summable and arithmetic convergent sequences, respectively, suggested by the geometric sum \(_G\sum_{k|m} f(k)\) as \(k\) ranges over the divisors of \(m\). We further obtain an analogous of Möbius inversion formula in the sense of geometric calculus and give interesting results in the geometric field.On \((\ell,m)\)-regular bipartition triples.https://zbmath.org/1449.050232021-01-08T12:24:00+00:00"Mahadeva Naika, M. S."https://zbmath.org/authors/?q=ai:mahadeva-naika.megadahalli-sidda"Nayaka, S. Shivaprasada"https://zbmath.org/authors/?q=ai:nayaka.s-shivaprasadaSummary: Let \(BT_{\ell,m}(n)\) denote the number of \((\ell,m)\)-regular bipartition triples of a positive integer \(n\). We establish infinite families of arithmetic identities and congruences for \(BT_{\ell,m}(n)\) modulo 3, 9 and 27 for various values of \(\ell\) and \(m\). For examples, we show that
\[
BT_{3,5}\left( 5^{\alpha +2}n+\frac{11\cdot 5^{\alpha +1}-3}{4}\right) \equiv 0 \pmod{9}
\]
and
\[
BT_{3,9}(27n+19)\equiv 0 \pmod{27}
\]
for all nonnegative integers \(\alpha\) and \(n\).A Diophantine inequality with two primes and one \(k\)-th power of a prime.https://zbmath.org/1449.110722021-01-08T12:24:00+00:00"Zhu, Li"https://zbmath.org/authors/?q=ai:zhu.liSummary: Let \(\psi (k) = \begin{cases}\frac{1}{2k}- \frac{3}{10}, & 1 < k < \frac{5}{4}, \\ \frac{1}{10}, & \frac{5}{4} \le k < 2, \\ \frac{2}{3k} - \frac{7}{30}, & 2 \le k \le \frac{5}{2}.\end{cases}\) Suppose that \({\lambda_1}, {\lambda_2}\) and \({\lambda_3}\) are non-zero real numbers, not all of the same sign, satisfying that \(\frac{\lambda_1}{\lambda_2}\) is irrational. Then for any real number \(\eta\) and \(\varepsilon > 0\), the inequality \[|{\lambda_1}{p_1} + {\lambda_2}{p_2} + {\lambda_3}p_3^k + \eta| \le (\max \{{p_1}, {p_2}, {p_3^k} \})^{-\psi (k) + \varepsilon}\] has infinitely many solutions in prime variables \({p_1}, {p_2}, {p_3}\).A note on primitive Heronian triangles.https://zbmath.org/1449.110192021-01-08T12:24:00+00:00"Zhang, Sihui"https://zbmath.org/authors/?q=ai:zhang.sihui"Jiang, Tianze"https://zbmath.org/authors/?q=ai:jiang.tianzeSummary: Infinitely many acute primitive Heronian triangles with integer inradius and exradii are obtained, and infinitely many primitive Heronian triangles with non-integer inradius and exradii are also obtained.On Pell hybrinomials.https://zbmath.org/1449.110322021-01-08T12:24:00+00:00"Liana, Mirosław"https://zbmath.org/authors/?q=ai:liana.miroslaw"Szynal-Liana, Anetta"https://zbmath.org/authors/?q=ai:szynal-liana.anetta"Włoch, Iwona"https://zbmath.org/authors/?q=ai:wloch.iwonaSummary: Hybrid numbers generalize complex, hyperbolic and dual numbers, simultaneously. Special kinds of hybrid numbers, related to numbers of Fibonacci type, among others Pell numbers, were introduced quite recently. In this paper we introduce and study polynomials, which are a generalization of Pell hybrid numbers and so called Pell hybrinomials.On the positive integer solution of Diophantine equation \(x (x+1) (x+2) (x+3) = 42y (y+1) (y+2) (y+3)\).https://zbmath.org/1449.110582021-01-08T12:24:00+00:00"Li, Jianglong"https://zbmath.org/authors/?q=ai:li.jianglong"Luo, Ming"https://zbmath.org/authors/?q=ai:luo.ming"Lin, Lijuan"https://zbmath.org/authors/?q=ai:lin.lijuanSummary: In this paper, \(M, N\) are both given positive integer, we study the resolution problem about the Diophantine equation
\[ Mx (x + 1) (x + 2) (x + 3) = Ny (y + 1) (y + 2) (y + 3). \]
Based on the basic solution of Pell equation, recursive sequence, congruence theory and other elementary methods, it is proved that when \( (M, N) = (1, 42)\), the Diophantine equation has only one positive integer solution, that is \( (x, y) = (7, 2)\). Furthermore, we prove that the equation has positive integer solution when \(M = 1\) and \(N \le 50\).Some identities involving Fibonacci sequence and Lucas sequence.https://zbmath.org/1449.050262021-01-08T12:24:00+00:00"Chen, Guohui"https://zbmath.org/authors/?q=ai:chen.guohuiSummary: The summation problem about the convolution sums of second-order linear recursion sequences is considered. According to the definition and properties of Fibonacci and Lucas polynomials, based on the existing research results, and by using the elementary method and the power series expansion of exponential function, some new calculation formulas of these second-order linear recursion sequence are obtained. In addition, by analyzing and generalizing these new results, a series of interesting identities are obtained.Power means of the Hurwitz zeta function over large intervals.https://zbmath.org/1449.110902021-01-08T12:24:00+00:00"Ashton, A. C. L."https://zbmath.org/authors/?q=ai:ashton.anthony-c-lSummary: In this note, we derive asymptotic formulas for power means of the Hurwitz zeta function over large intervals.New interesting Euler sums.https://zbmath.org/1449.110872021-01-08T12:24:00+00:00"Nimbran, Amrik Singh"https://zbmath.org/authors/?q=ai:nimbran.amrik-singh"Sofo, Anthony"https://zbmath.org/authors/?q=ai:sofo.anthonySummary: We present here some new and interesting Euler sums obtained by means of related integrals and elementary approach. We supplement Euler's general recurrence formula with two general formulas of the form
\[\sum_{n\geqslant 1} O_n^{(m)}\left(\frac{1}{(2n -1)^p} + \frac {1}{(2n)^p}\right)\quad\text{and}\quad\sum_{n\geqslant 1}\frac{O_n}{(2n-1)^p(2n+1)^q}, \]
where \(\displaystyle O_n^{(m)}= \sum_{j=1}^n \frac{1}{(2j-1)^m}\). Two formulas for \(\zeta (5)\) are also derived.Generalized Apostol-type polynomial matrix and its algebraic properties.https://zbmath.org/1449.110392021-01-08T12:24:00+00:00"Quintana, Yamilet"https://zbmath.org/authors/?q=ai:quintana.yamilet"Ramírez, William"https://zbmath.org/authors/?q=ai:ramirez.william"Urieles, Alejandro"https://zbmath.org/authors/?q=ai:urieles.alejandroThe authors introduce the concept of the generalized Apostol-type polynomial matrix and of the Apostol-type matrix which involve Apostol-type polynomials and their values. Here the generalized Apostol-type polynomials in the variable \(x\) and parameters \(c,a,\lambda,\mu,\nu\) order \(\alpha\) and level \(m\) are defined by the exponential generating function \(\big(E_{1,m+1}^{(c,a; \lambda;\mu;\nu)}(z)\big)^\alpha c^{xz}\) where \(E_{1,m+1}^{(c,a;\lambda;\mu;\nu)}(z)\) is the Mittag-Leffler type function. This class of polynomials has been introduced by \textit{P. Hernández-Llanos} et al. [Result. Math. 68, No. 1--2, 203--225 (2015; Zbl 1335.11014)] and provides a unified representation of the generalized Apostol-type polynomials and the generalized Apostol-Bernoulli polynomials, Apostol-Euler polynomials and Apostol-Genocchi polynomials. In the main results of the paper the authors prove a product formula for generalized Apostol-type polynomial matrices, show their relationship with generealized Pascal matrices of the first kind and provide some factorizations of the Apostol-type matrices in terms of the Fibonacci and Lucas matrices, respectively.
Reviewer: Štefan Porubský (Praha)Integral points on the elliptic curve \(y^2 = x^3 + (m - 4)x - 2m\).https://zbmath.org/1449.110562021-01-08T12:24:00+00:00"Guan, Xungui"https://zbmath.org/authors/?q=ai:guan.xunguiSummary: Let \(p, q\) be primes and \(m = 4p - 8 = q + 1\) or \(m = 2p - 8 = q + 1\) with \(p\not\equiv 1 \pmod 8\). We obtain all integral points \((x, y)\) on the elliptic curve \(y^2 = x^3 + (m - 4)x - 2m\).Hyper-atoms applied to the critical pair theory.https://zbmath.org/1449.110202021-01-08T12:24:00+00:00"Hamidoune, Yahya O."https://zbmath.org/authors/?q=ai:hamidoune.yahya-oThe sumset of two subsets \(A\) and \(B\) of an abelian group \(G\), is \(A+B=\{a+b \ | \ a\in A, b \in B \}.\) If \(A+B\) is aperiodic (that is the set has a trivial stabilizer), then it is known that \(|A+B| \geq |A|+|B| -1.\) \textit{J. H. B. Kemperman}'s theorem [Acta Math. 103, 63--88 (1960; Zbl 0108.25704)] describes the sets \(A\) and \(B\) for which \(|A+B| \leq |A|+|B| -1.\) Let \(\kappa_1(S) = \min \{ |X+S|-|S| \ | \ X\) is a finite subset of the group \(G\) with at least \(k\) elements satisfying \(|G \backslash (X+S)| \geq k \}.\) A maximal cardinality subgroup with \(\kappa_1(S) = |H+S| - |H|\) is a hyper-atom. The authors uses hyper-atoms to give a new isoperimetric proof of the Kemperman structure theorem which overcomes a previous weakness.
Reviewer: Steven T. Dougherty (Scranton)Menon-Sury's identity with several Dirichlet characters and additive characters.https://zbmath.org/1449.110022021-01-08T12:24:00+00:00"Chen, Man"https://zbmath.org/authors/?q=ai:chen.manSummary: This paper studies the Menon-Sury's identity with both Dirichlet characters and additive characters, and we shall give the explicit formula of the following sum
\[\sum_{\substack{a_1,\ldots, a_s\in \mathbb{Z}_n^\ast \\ b_1, \ldots, b_r\in \mathbb{Z}_n}} \gcd ({a_1}-1, \ldots, {a_s}-1, {b_1}, \ldots, {b_r}, n){\chi_1} ({a_1}) \cdots {\chi_s} ({a_s}){\lambda_1} ({b_1}) \cdots {\lambda_r} ({b_r}),\]
where \(n\) is a positive integer, \(s,r\) are nonnegative integers, \(\mathbb{Z}_n^*\) is the group of units of the ring \(\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z}\), \(\gcd (\cdot,\cdot)\) represents the greatest common divisor, \(\chi_i\) \((1 \le i \le s)\) are Dirichlet characters mod \(n\) with conductors \(d_i\), \(\lambda_j\) \((1 \le j \le r)\) are additive characters of \(\mathbb{Z}_n\). From the point of view of Fourier analysis on finite abelian groups, our result presents the explicit expression of Fourier coefficients of the function
\[ f(a_1, \ldots, a_s, b_1, \ldots, b_r) = \gcd (a_1-1, \ldots, a_s-1, b_1, \ldots, b_r, n)\]
on the abelian group \( (\mathbb{Z}_n^*)^s \times (\mathbb{Z}_n)^r\).Generalized Jacobsthal numbers and restricted \(k\)-ary words.https://zbmath.org/1449.050202021-01-08T12:24:00+00:00"Ramirez, José L."https://zbmath.org/authors/?q=ai:ramirez.jose-luis"Shattuck, Mark"https://zbmath.org/authors/?q=ai:shattuck.mark-aSummary: We consider a generalization of the problem of counting ternary words of a given length which was recently investigated by \textit{T. Koshy} and \textit{R. P. Grimaldi} [Fibonacci Q. 55, No. 2, 129--136 (2017; Zbl 1401.11042)]. In particular, we use finite automata and ordinary generating functions in deriving a \(k\)-ary generalization. This approach allows us to obtain a general setting in which to study this problem over a \(k\)-ary language. The corresponding class of \(n\)-letter \(k\)-ary words is seen to be equinumerous with the closed walks of length \(n-1\) on the complete graph for \(k\) vertices as well as a restricted subset of colored square-and-domino tilings of the same length. A further polynomial extension of the \(k\)-ary case is introduced and its basic properties deduced. As a consequence, one obtains some apparently new binomial-type identities via a combinatorial argument.On the largest part size and its multiplicity of a random integer partition.https://zbmath.org/1449.050242021-01-08T12:24:00+00:00"Mutafchiev, Ljuben"https://zbmath.org/authors/?q=ai:mutafchiev.lyuben-rSummary: Let \(\lambda\) be a partition of the positive integer \(n\) chosen uniformly at random among all such partitions. Let \(L_n = L_n(\lambda)\) and \(M_n = M_n(\lambda)\) be the largest part size and its multiplicity, respectively. For large \(n\), we focus on a comparison between the partition statistics \(L_n\) and \(L_n M_n\). In terms of convergence in distribution, we show that they behave in the same way. However, it turns out that the expectation of \(L_n M_n - L_n\) grows as fast as \(\frac{1}{2}\log n\). We obtain a precise asymptotic expansion for this expectation and conclude with an open problem arising from this study.On the character sums of polynomials and \(L\)-functions.https://zbmath.org/1449.110842021-01-08T12:24:00+00:00"Zhang, Jiafan"https://zbmath.org/authors/?q=ai:zhang.jiafan"Lv, Xingxing"https://zbmath.org/authors/?q=ai:lv.xingxingSummary: The main purpose of this paper is using the analytic methods and the properties of character sums to study the computational problem of one kind hybrid power mean involving the character sums of polynomials and Dirichlet \(L\)-functions modulo \(p\), an odd prime, and give some exact computational formulas for them.Note on the Steinhaus problem.https://zbmath.org/1449.110752021-01-08T12:24:00+00:00"Guan, Xungui"https://zbmath.org/authors/?q=ai:guan.xunguiSummary: In the paper, by pointing out some errors in other articles, we knew that the well known integer distance point problem of Steinhaus was still open. Using the property of Pythagoras triple and the infinite descent method, some statements concerning the nonexistence of Steinhaus points were obtained, and an open problem given by other articles was partly solved.An iterative method based on Pohlig-Hellman and Pollard \(\rho \) for computing discrete logarithm.https://zbmath.org/1449.111142021-01-08T12:24:00+00:00"Hu, Jianjun"https://zbmath.org/authors/?q=ai:hu.jianjun"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.30"Li, Hengjie"https://zbmath.org/authors/?q=ai:li.hengjieSummary: The advantage of the Pohlig-Hellman algorithm is that the calculation speed is fast, but the disadvantage is that the order of the group requires to be smooth. The advantage of Pollard \(\rho \) algorithm is that it is not limited to the group structure, but the disadvantage is the probability algorithm, and the accuracy of calculation is usually lower than the Pohlig-Hellman algorithm. Scholars paid little attention to effective fusion of both Pollard \(\rho \) and Pohlig-Hellman algorithms. In order to solve the problem, combined with the respective strengths of the two algorithms, the paper put forward a kind of Pollard \(\rho \) mixed discrete logarithm iteration algorithm based on Pohlig-Hellman. The idea of the algorithm was that the Pohlig-Hellman algorithm was used to iterate when the order factor was less than or equal to the smooth boundary, and the Pollard \(\rho \) algorithm was used to iterate when the order factor was greater than the smooth boundary. At the same time, the computational efficiency of the hybrid algorithm was analyzed, and the correctness and validity of the algorithm and the analysis results were verified by an example.Solutions of two equations related to generalized Euler function.https://zbmath.org/1449.110132021-01-08T12:24:00+00:00"Zhang, Sibao"https://zbmath.org/authors/?q=ai:zhang.sibaoSummary: Let \({\varphi_e} (n)\) be generalized Euler function, where \(n\) and \(e\) are positive integers. The positive integer solutions of two equations \({\varphi_3} (n) = {2^{\omega (n)}}\) and \({\varphi_4} (n) = {2^{\omega (n)}}\) with generalized Euler function were studied. Based on the properties of generalized Euler function and Euler function, all solutions of the two equations were obtained by using the mode of discussion on classification and segmentation, where \(\omega (n)\) was the number of the different prime factor of positive integer \(n\).Solutions to a class of equations in integral matrices of order 2.https://zbmath.org/1449.150442021-01-08T12:24:00+00:00"Yin, Qianqian"https://zbmath.org/authors/?q=ai:yin.qianqian"Liang, Xinran"https://zbmath.org/authors/?q=ai:liang.xinran"Yuan, Pingzhi"https://zbmath.org/authors/?q=ai:yuan.pingzhiSummary: To solve the problems of integer matrix equations related to Pythagorean equation \({x^2} + {y^2} = {z^2}\), the solutions \( (X, Y)\) to the \(2 \times 2\) integral matrix equation \(X^2 + Y^2 = \lambda I\), where \(\lambda \in \mathbb{Z}\) and \(I\) is the unit matrix, which are related to the Pythagorean equation, are investigated and completely solved by using the basic operation of matrix to transform the problem of integer matrix equation into the problem of solving some Diophantine equations, which is gradually extended from the special case to the general case. The solutions to \(2 \times 2\) integral matrix equation \(X^2 - Y^2 = \lambda I\) also can be solved with similar methods.A remark on bounds for the maximal height of divisors of \(x^n-1\).https://zbmath.org/1449.110492021-01-08T12:24:00+00:00"Zhang, Bin"https://zbmath.org/authors/?q=ai:zhang.bin.2|zhang.bin.3|zhang.bin.1|zhang.bin.4Given a positive integer \(n\), let \(B(N)\) denote the maximum absolute value of coefficients of any divisor of polynomial \(x^n-1\). \textit{C. Pomerance} and \textit{N. C. Ryan} [Ill.\ J.\ Math. 51, No. 2, 597--604 (2007; Zbl 1211.11108)] proved that \(B(p^n)=1\) and that \(B(p_1p_2)=\min(p_1,p_2)\) and conjectured that \(B(p_1^2p_2)=\min(p_1^2,p_2)\), where \(p_1,p_2\) are distinct primes. \textit{N. Kaplan} [J. Number Theory 129, No. 11, 2673--2688 (2009; Zbl 1250.11028)] proved this conjecture and showed \((3p_1^2p_2-p_1^3+7p_1-6)/3\leq B(p_1p_2p_3)\leq p_1^2p_2^2\) where \(p_1<p_2<p_3\) that \(B(n)\) can be bounded by a function that does not depend on the largest prime factor of \(n\). In the present paper the author proves that \(B(p_1^{e_1}\dots p_r^{e_r})=(2/5)^C\prod_{i=1}^{r-1} p_i^{4\cdot 3^{r-2}E-e_i}\), where \(C=\prod_{i=2}^{r}e_i\), \(E=\prod_{j=1}^{r}e_j\) and \(p_1<\dots<p_r\), \(r\ge 2\).
Reviewer: Štefan Porubský (Praha)Generalized dominoes tiling's Markov chain mixes fast.https://zbmath.org/1449.050462021-01-08T12:24:00+00:00"Kayibi, K. K."https://zbmath.org/authors/?q=ai:kayibi.koko-kalambay"Samee, U."https://zbmath.org/authors/?q=ai:samee.umatul|samee.uma"Merajuddin"https://zbmath.org/authors/?q=ai:merajuddin.pirzada|merajuddin.m"Pirzada, S."https://zbmath.org/authors/?q=ai:pirzada.shariefuddinSummary: A generalized tiling is defined as a generalization of the properties of tiling a region of \(\mathbb{Z}^2\) with dominoes, and comprises tiling with rhombus and any other tilings that admits height functions which can be ordered into a distributive lattice. By using properties of the distributive lattice, we prove that the Markov chain consisting of moving from one height function to the next by a flip is fast mixing and the mixing time \(\tau(\epsilon)\) is given by \(\tau(\epsilon)\leq(kmn)^3(mn\;\ln\;k+\ln\;\epsilon^{-1})\), where \(mn\) is the area of the grid \(\Gamma\) that is a \(k\)-regular polycell. This result generalizes the result of \textit{K. K. Kayibi} and \textit{S. Pirzada} [Theor. Comput. Sci. 714, 1--14 (2018; Zbl 1387.05043)] and improves on the mixing time obtained by using coupling arguments by \textit{N. Destainville} [in: Discrete models: combinatorics, computation, and geometry. Proceedings of the 1st international conference (DM-CCG), Paris, France, July 2--5, 2001. Paris: Maison de l'Informatique et des Mathématiques Discrètes (MIMD). 1--22 (2001; Zbl 1017.68148)] and by \textit{M. Luby} et al. [SIAM J. Comput. 31, No. 1, 167--192 (2001; Zbl 0992.82013)].On the Riemann Zeta function, I: KS-transform.https://zbmath.org/1449.110892021-01-08T12:24:00+00:00"Ge, Liming"https://zbmath.org/authors/?q=ai:ge.limingSummary: Kadison-Singer transform (KS-transform) is introduced as a multiplicative Fourier transform associated with the multiplicative structure of natural numbers. It is a unitary operator between the Hilbert space \({L^2} ([1, \infty))\) and Hardy space \({H^2} (\Omega)\), where \(\Omega\) is a the right half complex plane with the real part great than or equal to 1/2. We also show that KS-transform maps the multiplicative convolution of two functions on \([1, \infty)\) to the usual product of functions on \(\Omega\). Riemann hypothesis is equivalent to the vanishing index of certain convolution operators.Application of quartic residue character theory to the Diophantine equation \(a^x+b^y=c^z\).https://zbmath.org/1449.110642021-01-08T12:24:00+00:00"Deng, Mou-Jie"https://zbmath.org/authors/?q=ai:deng.moujie"Guo, Jin"https://zbmath.org/authors/?q=ai:guo.jinLet \((a,b,c)\) be a primitive Pythagorian triple, represented as usually as \(a=m^2-n^2, b=2mn, c=m^2+n^2\) with coprime positive integers \(m,n\) of distinct parity. Jeśmanovic conjectured that \(a^x+b^y=c^z\) can only be satisfied by \(x=y=z=2\). The conjecture has been proved formerly in the case \(2\Vert mn\). The present paper is a new contribution to the literature of this topic. The authors prove that the conjecture is true if \(m\equiv 4\) (mod 8), \(n\equiv 9\) (mod 16) or if \(m\equiv 9\) (mod 16), \(n\equiv 4\) (mod 8). The proofs are based on the theory of quartic residue characters, elementary methods and a result of \textit{T. Miyazaki} [Bull. Aust. Math. Soc. 80, No. 3, 413--422 (2009; Zbl 1225.11038)].
Reviewer: István Gaál (Debrecen)On a certain class of arithmetic functions.https://zbmath.org/1449.110082021-01-08T12:24:00+00:00"Oller-Marcén, Antonio M."https://zbmath.org/authors/?q=ai:oller-marcen.antonio-mLet \(R\) denote a ring. An arithmetic function \(f:\mathbb{N}\to R\) is called homothetic of ratio \(K\) if \(f(Kn)=f(n)\) for every \(n\in\mathbb{N}\). If \(f\) is periodic, then \(f\) is homothetic. More exactly, if \(f\) is periodic of period \(T\), then \(f\) is homothetic of ratio \(K\), where \(K\) is any integer such that \(K\equiv 1\) (mod \(T\)). The converse is not true.
The paper under review gives an upper bound for the number of elements of \(f(\mathbb{N})\), with \(f\) periodic, in terms of the period and ratio of \(f\). The main result is the following Theorem: Let \(f:\mathbb{N}\to R\) be a periodic function of period \(T\) which is also homothetic of ratio \(K\). Let \(T=p_1^{r_1}\cdots p_s^{r_s}\). Then \[\# f(\mathbb{N})\le\prod_{p_i\nmid K}\left(\frac{p_i^{r_i-1}(p_i-1)}{{\operatorname{ord}}_{p_i^{r_i}}(K)}+p_i^{r_i-1}\right),\] where \({\operatorname{ord}}_{p_i^{r_i}}(K)\) is the order of \(K\) (mod \(p_i^{r_i}\)).
Reviewer: László Tóth (Pécs)On the height of cyclotomic polynomials.https://zbmath.org/1449.110422021-01-08T12:24:00+00:00"Al-Kateeb, Alaa"https://zbmath.org/authors/?q=ai:al-kateeb.alaaLet \(\Phi_m(X)\) be the \(m\)th cyclotomic polynomial, denote by \(C(\Phi_m)\) the set of its coefficients and by \(h(\Phi_m)\) its height, i.e., \[h(\Phi_m)=\max\{\vert a\vert:\ a\in C(\Phi_m)\}.\] The author shows that if \(m>0\) is an odd square-free integer, \(p>m\) is a prime and \(p\equiv r\) mod \(m\) with either \(0<r<2m/(\varphi(m)-2)\) or \(p>r>m-2m/ (\varphi(m)-2)\), then \[h(\Phi_{pm})\ge h(\Phi_m).\] The proof is based on certain properties of cyclotomic polynomials established in a paper by the author et al. [``Block structure of cyclotomic polynomials'', \url{arXiv:1704.04051}], and uses the result of \textit{N. Kaplan} [Integers 10, A30, 357--363 (2010; Zbl 1200.11020)] showing that if \(m\) is odd and square-free and for primes \(p,q>m\) one has \(p\equiv q\) mod \(m\), then \(C(\Phi_{pm})=C(\Phi_{qm})\).
Reviewer: Władysław Narkiewicz (Wrocław)\(m\) extension of Lucas \(p\)-numbers in information theory.https://zbmath.org/1449.110352021-01-08T12:24:00+00:00"Prasad, Bandhu"https://zbmath.org/authors/?q=ai:prasad.bandhuSummary: In this paper, we introduced a new Lucas \(Q_{p,m}\) matrix for \(m\)-extension of Lucas \(p\)-numbers where \(p\) (\(> 0\)) is integer and \(m\) (\(> 0\)). Thereby, we discuss various properties of \(Q_{p,m}\) matrix, coding and decoding theory followed from the \(Q_{p,m}\) matrix.A generalization of André-Jeannin's symmetric identity.https://zbmath.org/1449.050272021-01-08T12:24:00+00:00"Munarini, Emanuele"https://zbmath.org/authors/?q=ai:munarini.emanueleSummary: \textit{R. André-Jeannin} [Fibonacci Q. 35, No. 1, 68--74 (1997; Zbl 0879.11006)] obtained a symmetric identity involving the reciprocal of the Horadam numbers \(W_n\), defined by a three-term recurrence \(W_{n+2}= PW_{n+1} - QW_n\) with constant coefficients. In this paper, we extend this identity to sequences \(\{a_n\}_{n\in N}\) satisfying a three-term recurrence \(a_{n+2} = p_{n+1} a_{n+1} + q_{n+1} a_n\) with arbitrary coefficients. Then, we specialize such an identity to several \(q\)-polynomials of combinatorial interest, such as the \(q\)-Fibonacci, \(q\)-Lucas, \(q\)-Pell, \(q\)-Jacobsthal, \(q\)-Chebyshev and \(q\)-Morgan-Voyce polynomials.The integer solution of the Diophantine equation \(x^2 + 4^n = y^9\).https://zbmath.org/1449.110602021-01-08T12:24:00+00:00"You, Lihua"https://zbmath.org/authors/?q=ai:you.lihua"Cai, Xiaoqun"https://zbmath.org/authors/?q=ai:cai.xiaoqunSummary: It is proved that the Diophantine equation \(x^2 + 4^n = y^9\) has no integer solution, where \(x \equiv 1\pmod 2\). It is further proved that the Diophantine equation \(x^2 + 4^n = y^9\) \((n = 6, 7, 8)\) has no integer solution. Then it is shown that the Diophantine equation \(x^2 + 4^n = y^9\) has integer solution if and only if \(n \equiv 4\pmod 9\), and \( (x, y) = (0, {4^m})\) when \(n = 9m\) or \( (x, y) = (\pm {2^{9m + 4}}, {2^{2m + 1}})\) when \(n = 9m + 4\), where \(m \in N\).
Furthermore, based on the results of \(k = 5, 9\), a conjecture about the integer solutions of the Diophantine equation \(x^2 + 4^n = y^k\) for further research is proposed, where \(k\) is odd.The square mapping graph of \(2\times 2\) matrix rings over prime fields.https://zbmath.org/1449.051392021-01-08T12:24:00+00:00"Tang, Gaohua"https://zbmath.org/authors/?q=ai:tang.gaohua"Zhang, Hengbin"https://zbmath.org/authors/?q=ai:zhang.hengbin"Wei, Yangjiang"https://zbmath.org/authors/?q=ai:wei.yangjiangSummary: Let \(R\) be a ring with identity. The square mapping graph of \(R\) is a digraph \(\Gamma(R)\) defined on the elements of \(R\) and with an edge from a vertex a to \(a\) vertex \(b\) if and only if \(a^2=b\). Let \(\mathbb{M}_2(\mathbb{Z}_p)\) be the \(2\times 2\) matrix ring over the field \(\mathbb{Z}_p\), where \(p\) is prime. In this paper, we completely determine the structure of \(\Gamma(\mathbb{M}_2(\mathbb{Z}_p))\) by showing its two disjoint induced subgraphs.On additive arithmetical functions with values in topological groups. IV.https://zbmath.org/1449.110992021-01-08T12:24:00+00:00"Kátai, Imre"https://zbmath.org/authors/?q=ai:katai.imre"Phong, Bui Minh"https://zbmath.org/authors/?q=ai:bui-minh-phong.Let \(\varphi,\psi: \mathbb{N} \to G\) two completely additive functions, where \(G\) is an abelian topological group. The authors provide two conditions, each of them implying that \(\varphi=\psi\). They also deal with the extensions of the functions \(\varphi\) and \(\psi\) to the multiplicative group of positive real numbers.
For Part III, see Publ. Math. 94, No. 1-2, 49--54 (2019; Zbl 1424.11144).
Reviewer: Moshe Roitman (Haifa)On one property of the weighed sums of equal powers as matrix products.https://zbmath.org/1449.110692021-01-08T12:24:00+00:00"Nikonov, Aleksandr Ivanovich"https://zbmath.org/authors/?q=ai:nikonov.aleksandr-ivanovichSummary: Finite sum of weighted sum of equal powers is represented in matrix form. This is expressed by the presence of two matrix components, the first of which does not contain, and the second contains the specified weights. It is significant that if the maximum power grounds in the amounts of this type has a value of no less important exponent related to its term, the first matrix component is the same for all amounts that meet the specified condition. Revealed common property of the matrix representation used to obtain the modified final sum from the increased number of weights.On binomial double sums with Fibonacci and Lucas numbers. II.https://zbmath.org/1449.110312021-01-08T12:24:00+00:00"Kılıç, Emrah"https://zbmath.org/authors/?q=ai:kilic.emrah"Taşdemir, Funda"https://zbmath.org/authors/?q=ai:tasdemir.fundaSummary: In this paper, we compute various binomial-double-sums involving the Fibonacci numbers as well as their alternating analogous. It would be interesting that all sums we shall compute are evaluated in nice multiplication forms in terms of again the Fibonacci and Lucas numbers.
For Part I, see Ars Comb. 144, 173--185 (2019; Zbl 07144788).Bicomplex generalized \(k\)-Horadam quaternions.https://zbmath.org/1449.110372021-01-08T12:24:00+00:00"Yazlik, Yasin"https://zbmath.org/authors/?q=ai:yazlik.yasin"Köme, Sure"https://zbmath.org/authors/?q=ai:kome.sure"Köme, Cahit"https://zbmath.org/authors/?q=ai:kome.cahitSummary: This study provides a broad overview of the generalization of the various quaternions, especially in the context of its enhancing importance in the disciplines of mathematics and physics. By the help of bicomplex numbers, in this paper, we define the bicomplex generalized \(k-\)Horadam quaternions. Fundamental properties and mathematical preliminaries of these quaternions are outlined. Finally, we give some basic conjucation identities, generating function, the Binet formula, summation formula, matrix representation and a generalized identity, which is generalization of the well-known identities such as Catalan's identity, Cassini's identity and d'Ocagne's identity, of the bicomplex generalized \(k-\)Horadam quaternions in detail.On some consequences of recently proved conjectures.https://zbmath.org/1449.110982021-01-08T12:24:00+00:00"De Koninck, Jean-Marie"https://zbmath.org/authors/?q=ai:de-koninck.jean-marie"Kátai, Imre"https://zbmath.org/authors/?q=ai:katai.imre"Phong, Bui Minh"https://zbmath.org/authors/?q=ai:bui-minh-phong.Let \(\mathcal{A}\) stand for the set of real-valued additive function, \(\triangle h(n)=h(n+1)-h(n)\) for a given \(h\in\mathcal{A}\), and \(\Vert \cdot\Vert \) for the nearest integer function. The authors provide and update some results connected with a cluster of conjectures related to \textit{I. Kátai}'s ones [Stud. Sci. Math. Hung. 16, 289--295 (1981; Zbl 0479.10003)] based on the following main result of the paper: Let \(h\in\mathcal{A}\) and \(\tau\) be an irrational number. If \(\lim_{x\to\infty}\omega(x)\sum_{n\le x}\Vert \triangle h(n)\Vert =0\) and \(\lim_{x\to\infty}\omega(x)\Vert \tau\triangle h(n)\Vert =0\) where in both identities simultaneously either \(\omega(x)=x^{-1}\) or \(\omega(x)=(x\log x)^{-1}\) then there exists a real \(c\) such that \(h(n)=c\log x\) for all positive integers \(n\).
Reviewer: Štefan Porubský (Praha)\(p\)-adic BMO and VMO functions.https://zbmath.org/1449.110732021-01-08T12:24:00+00:00"Zelenov, Evgeniĭ Igor'evich"https://zbmath.org/authors/?q=ai:zelenov.evgenii-iSummary: Spaces of \(p\)-adic BMO and VMO functions are considered. It is proved that locally constant functions are dense in VMO space under BMO norm.On two bivariate kinds of \((p,q)\)-Bernoulli polynomials.https://zbmath.org/1449.330162021-01-08T12:24:00+00:00"Sadjang, P. Njionou"https://zbmath.org/authors/?q=ai:sadjang.p-njionou|sadjang.patrick-njionou"Duran, Ugur"https://zbmath.org/authors/?q=ai:duran.ugurSummary: The main aim of this paper is to introduce and investigate \((p,q)\)-extensions of two bivariate kinds of Bernoulli polynomials and numbers. We firstly examine several \((p,q)\)-analogues of the Taylor expansions of products of some trigonometric functions and determine their coefficients which are also analyzed in detail. Then, we introduce two bivariate kinds of \((p,q)\)-Bernoulli polynomials and acquired multifarious formulas and relations including connection formulas, recurrence formulas, correlations with aforementioned coefficients, partial \((p,q)\)-differential equations and \((p,q)\)-integral representations.Special forms and the distribution of practical numbers.https://zbmath.org/1449.110962021-01-08T12:24:00+00:00"Wu, X.-H."https://zbmath.org/authors/?q=ai:wu.xianhong|wu.xiaohe|wu.xionghua|wu.xuehong|wu.xiuhen|wu.xinghua|wu.xihui|wu.xianghui|wu.xuhong|wu.xiaohui|wu.xihuan|wu.xuanhui|wu.xiuheng|wu.xuehui|wu.xiaohua|wu.xiaohuan|wu.xinhui|wu.xianghong|wu.xianghua|wu.xiuhua|wu.xiaohong|wu.xiaohan|wu.xiaohang\textit{A. K. Srinivasan} [Practical numbers. Curr. Sci. 6, 179--180 (1948)] introduced the notion of practical numbers (a positive integer \(n\) is said to be practical if every positive integer \(m\leqslant n\) can be written as a sum of distinct divisors of \(n\)) and showed that a practical number greater than \(2\) is divisible by \(4\) or \(6\). \textit{B. M. Stewart} [Am. J. Math. 76, 779--785 (1954). Zbl 0056.27004] gave a characterization of practical numbers in terms of their prime factorization, and this is used here to show two results on practical numbers as follows. For positive integers \(a,b,k\) with \(a\) odd, it is shown that \(am^k+bm^{k-1}\) is practical for infinitely many \(m\geqslant1\). The second result shows that for \(n\geqslant7\) there are at least two practical numbers in the interval \((n^2,(n+1)^2)\). It is also conjectured that for any \(k\) this interval contains at least \(k\) practical numbers for \(n\) large enough.
Reviewer: Thomas B. Ward (Leeds)A sharp upper bound for the sum of reciprocals of least common multiples.https://zbmath.org/1449.110032021-01-08T12:24:00+00:00"Hong, S. A."https://zbmath.org/authors/?q=ai:hong.siaoLet \((a_i)\) be a strictly increasing sequence of positive integers. Then \textit{D. Borwein} [Can. Math. Bull. 21, 117--118 (1978; Zbl 0376.10003)] proved, confirming a conjecture of Erdős, that \[ \sum_{i=1}^{n-1}\frac1{\text{lcm}(a_i,a_{i+1})}\le1-\frac1{2^{n-1}}, \] with equality holding if and only if \(a_i=2^{i-1}\). The author of the paper under review considers a generalization of this result, namely the problem of estimating the sums \[ S_{n,k}=\sum_{i=1}^{n-k}\frac1{\text{lcm}(a_i,a_{i+k})}. \] For \(k=2, 3\), he obtains optimal upper bounds for \(S_{n,k}\), as well as characterizations of the corresponding sequences \((a_i)\). These results comprise too many cases to reproduce here, but, by way of a sample result, we state the simplest estimate \[ S_{2m,2}\le\frac76\Bigl(1-\frac1{2^{m-1}}\Bigr). \]
Reviewer: Gennady Bachman (Las Vegas)\(x\)-Coordinates of Pell equations which are Tribonacci numbers. II.https://zbmath.org/1449.110302021-01-08T12:24:00+00:00"Kafle, Bir"https://zbmath.org/authors/?q=ai:kafle.bir"Luca, Florian"https://zbmath.org/authors/?q=ai:luca.florian"Togbé, Alain"https://zbmath.org/authors/?q=ai:togbe.alainLet \(d\ge 2\) be square-free. Let \((x_n, y_n)\) be positive integer solutions to the Pell equation \(x^2-dy^2=\pm 4\). The Tribonacci numbers \(T_m\) are defined by the recurrence \(T_{m+3}=T_{m+2}+T_{m+1}+T_m\) \((m\ge 0)\) with initial conditions \(T_0=0\) and \(T_1=T_2=1\). The authors prove that the Diophantine equation \(x_n=T_m\) has at most one solution \((n, m)\) in positive integers except for \(d=5\). The case \(d=5\) is completely characterized separately. This result is similar to that for the Pell equation \(x^2-dy^2=\pm 1\) considered in \textit{F. Luca} [Acta Arith. 179, No. 1, 25--35 (2017; Zbl 1410.11007)].
Reviewer: Pentti Haukkanen (Tampere)An additive equation involving fractional powers.https://zbmath.org/1449.111022021-01-08T12:24:00+00:00"Zhu, L."https://zbmath.org/authors/?q=ai:zhu.linfu|zhu.laiyu|zhu.lipin|zhu.luning|zhu.longfei|zhu.lilong|zhu.long|zhu.lirong|zhu.liling|zhu.liqin|zhu.liangyu|zhu.lidong|zhu.linfa|zhu.linjie|zhu.liu|zhu.licai|zhu.liya|zhu.lina|zhu.lemin|zhu.longchao|zhu.lingxiang|zhu.limin|zhu.linlin|zhu.lihua|zhu.liang|zhu.lian|zhu.liangjia|zhu.lin|zhu.liao|zhu.leqing|zhu.linhu|zhu.lierong|zhu.liqun|zhu.linbo|zhu.lijing|zhu.lianhua|zhu.liangkuan|zhu.linli|zhu.laiyi|zhu.luoding|zhu.lingfeng|zhu.libing|zhu.lizhong|zhu.liangliang|zhu.longyin|zhu.lei|zhu.lili|zhu.lingzhi|zhu.lanping|zhu.liangshun|zhu.lixing|zhu.lingjiong|zhu.luchuang|zhu.lingyan|zhu.lulu|zhu.lifeng|zhu.lanjian|zhu.lianqing|zhu.liping|zhu.lemei|zhu.lanying|zhu.lianyan|zhu.lianfang|zhu.linling|zhu.lutao|zhu.lianhong|zhu.liwen|zhu.lijie|zhu.lianxuan|zhu.ling|zhu.lailai|zhu.leilei|zhu.liusan|zhu.liye|zhu.longjie|zhu.lizhi|zhu.liqing|zhu.liguang|zhu.leqi|zhu.liying|zhu.lixia|zhu.lie|zhu.lifei|zhu.lijuan|zhu.lingjiang|zhu.lisha|zhu.lihong|zhu.lianning|zhu.lu|zhu.lisa|zhu.langfeng|zhu.linsheng|zhu.litao|zhu.liqiang|zhu.lujin|zhu.lingyun|zhu.lengxue|zhu.lingmei|zhu.linhe|zhu.lingshan|zhu.liyun|zhu.li|zhu.lijun|zhu.liyong|zhu.limei|zhu.lintao|zhu.lan|zhu.lichun|zhu.liuhai|zhu.liting|zhu.liehuang|zhu.lianghui|zhu.lingxue|zhu.ligeng|zhu.liuhua|zhu.lida|zhu.liyanLet \([x]\) the integer part of the real number \(x\). In this paper the author deals with the equation \([p_1^c]+[p_2^c]=n\), where \(p_1\) and \(p_2\) are primes. He shows that, for any \(c\in (1,14/11)\) and all \(n\in (N/2,N]\) but \(O(N\exp{(-\log^1/6{N})})\) exceptions, this equation is solvable. Let \(\mathcal{R}(n)=\sum_{[p_1^c]+[p_2^c]=n}\log{p_1}\log{p_2}\). He also establishes that, for the same values of \(n\), \(\mathcal{R}(n)=\frac{\Gamma^2(\frac{1}{c}+1)}{\Gamma(\frac{2}{c})} n^{\frac{2}{c}-1} +(N^{\frac{2}{c}-1}\exp{(-\log^{1/6}{N})})\). These results generalize some previously known observations from this area of number theory.
Reviewer: Ljuben Mutafchiev (Sofia)New reciprocal sums involving finite products of second order recursions.https://zbmath.org/1449.110232021-01-08T12:24:00+00:00"Kılıç, Emrah"https://zbmath.org/authors/?q=ai:kilic.emrah"Ersanlı, Didem"https://zbmath.org/authors/?q=ai:ersanli.didemSummary: In this paper, we present new kinds of reciprocal sums of finite products of general second order linear recurrences. In order to evaluate explicitly them by \(q\)-calculus, first we convert them into their \(q\)-notation and then use the methods of partial fraction decomposition and creative telescoping.Sums of four prime cubes in short intervals.https://zbmath.org/1449.111002021-01-08T12:24:00+00:00"Languasco, A."https://zbmath.org/authors/?q=ai:languasco.alessandro"Zaccagnini, A."https://zbmath.org/authors/?q=ai:zaccagnini.alessandroSummary: We prove that a suitable asymptotic formula for the average number of representations of integers \(n = p_1^3+p_2^3+p_3^3+p_4^3\), where \(p_1,p_2,p_3,p_4\) are prime numbers, holds in intervals shorter than the the ones previously known.On several permutation polynomials over \({F_{q^2}}\).https://zbmath.org/1449.111122021-01-08T12:24:00+00:00"Zhou, Fangmin"https://zbmath.org/authors/?q=ai:zhou.fangminSummary: Let \({F_{q^2}}\) be finite fields with \({q^2}\) elements. In this paper, we discuss the necessary and sufficient condition for \(x (1+ tx^{2 (q-1)}), t \in {F_{q^2}^*}\) to be permutation polynomials over \({F_{q^2}}\), and \(N (x) = x^{1+q}, x (s+tN (x)+N (x)^2), s,t \in {F_q}\) to be permutation polynomials over \({F_{q^2}}\).On the extensibility of \(D(-1)\)-pairs containing Fermat primes.https://zbmath.org/1449.110532021-01-08T12:24:00+00:00"Jukić, Bokun M."https://zbmath.org/authors/?q=ai:jukic.bokun-m"Soldo, I."https://zbmath.org/authors/?q=ai:soldo.ivanSummary: We study the extendibility of a \(D(-1)\)-pair \(\{1, p\}\), where \(p\) is a Fermat prime, to a \(D(-1)\)-quadruple in \(\mathbb{Z}[\sqrt{-t}], t > 0\).Elliptic curve scalar multiplication algorithm based on bronze ratio addition chain.https://zbmath.org/1449.940712021-01-08T12:24:00+00:00"Liu, Shuanggen"https://zbmath.org/authors/?q=ai:liu.shuanggen"Li, Dandan"https://zbmath.org/authors/?q=ai:li.dandan"Li, Xiao"https://zbmath.org/authors/?q=ai:li.xiaoSummary: A new efficient and secure elliptic curve scalar multiplication algorithm is proposed. There is a new addition chain based on generalized Fibonacci sequences, which is called bronze ratio addition chain (BRAC). Each iteration of this algorithm executes fixed ``\(3{P_1} + {P_2}\)'' operation, which can resist the simple power analysis naturally. BRAC has a shorter chain length, and can improve the efficiency of the previous ones by combining with the new projection coordinates. The experimental results show that the new algorithm is 31.73\% faster than golden ratio addition chain.Solving Pell equations via reduced regular continued fractions.https://zbmath.org/1449.110522021-01-08T12:24:00+00:00"Gao, Sheng"https://zbmath.org/authors/?q=ai:gao.shengSummary: In this paper, the author extends a well-known identity on periodic simple continued fractions to the case of reduced regular continued fractions, thereby representing all the solutions of one class of Pell equations via the convergence of reduced regular continued fractions.On the monoid of monic binary quadratic forms.https://zbmath.org/1449.110972021-01-08T12:24:00+00:00"Dimabayao, Jerome T."https://zbmath.org/authors/?q=ai:dimabayao.jerome-tomagan"Ponomarenko, Vadim"https://zbmath.org/authors/?q=ai:ponomarenko.vadim"Tigas, Orland James Q."https://zbmath.org/authors/?q=ai:tigas.orland-james-qSummary: We consider the quadratic form \(x^2+mxy+ny^2\), where \(\vert m^2-4n\vert \) is a prime number. Under the assumption that a particular, small, finite set of integers is representable, we determine all integers representable by this quadratic form.The explicit formula for a special class of generalized Euler functions.https://zbmath.org/1449.110072021-01-08T12:24:00+00:00"Liao, Qunying"https://zbmath.org/authors/?q=ai:liao.qunyingSummary: For a fixed positive integer \(n\), in order to generalize the modulo from the square of prime numbers to the square of an arbitrary integer for the well known Lehmer congruence formula, some previous researchers defined the generalized Euler function \(\varphi_e(n)\) in 2007 and then determined the explicit formulas for \(\varphi_e(n)\) \((e = 3, 4, 6)\) in 2013 and 2016. The present paper continues the study, obtains the computational formula of \(\varphi_e(n)\) for some special divisor \(e\) of \(n\), which is a generalization for the corresponding results of above, and then gives a sufficient and necessary condition for \(2\mid \varphi_e(n)\).On the generalized bi-periodic Fibonacci and Lucas quaternions.https://zbmath.org/1449.110272021-01-08T12:24:00+00:00"Choo, Younseok"https://zbmath.org/authors/?q=ai:choo.younseokSummary: In this paper we introduce the generalized bi-periodic Fibonacci and Lucas quaternions which are the further generalizations of the bi-periodic Fibonacci and Lucas quaternions considered in the literature. For those quaternions, we derive the generating functions, Binet's formulas and Catalan's identities.A family of lacunary recurrences for Fibonacci numbers.https://zbmath.org/1449.110242021-01-08T12:24:00+00:00"Ballantine, Cristina"https://zbmath.org/authors/?q=ai:ballantine.cristina-m"Merca, Mircea"https://zbmath.org/authors/?q=ai:merca.mirceaSummary: We introduce an infinite family of lacunary recurrences for the Fibonacci numbers and give a combinatorial proof. The first entry in the family was proved by Lucas in 1876.On a system of difference equations of second order solved in closed form.https://zbmath.org/1449.390032021-01-08T12:24:00+00:00"Akrour, Youssouf"https://zbmath.org/authors/?q=ai:akrour.youssouf"Touafek, Nouressadat"https://zbmath.org/authors/?q=ai:touafek.nouressadat"Halim, Yacine"https://zbmath.org/authors/?q=ai:halim.yacineSummary: In this work we solve in closed form the system of difference equations \[ x_{n+1}=\dfrac{ay_nx_{n-1}+bx_{n-1}+c}{y_nx_{n-1}},\; y_{n+1}=\dfrac{ax_ny_{n-1}+by_{n-1}+c}{x_ny_{n-1}},\;n=0,1,\dots,\] where the initial values \(x_{-1}\), \(x_0\), \(y_{-1}\) and \(y_0\) are arbitrary nonzero real numbers and the parameters \(a\), \(b\) and \(c\) are arbitrary real numbers with \(c\ne 0\). In particular we represent the solutions of some particular cases of this system in terms of Tribonacci and Padovan numbers and we prove the global stability of the corresponding positive equilibrium points. The results obtained here extend those obtained in some recent papers.An exponential Diophantine equation related to the difference between powers of two consecutive Balancing numbers.https://zbmath.org/1449.110362021-01-08T12:24:00+00:00"Rihane, Salah Eddine"https://zbmath.org/authors/?q=ai:rihane.salah-eddine"Faye, Bernadette"https://zbmath.org/authors/?q=ai:faye.bernadette"Luca, Florian"https://zbmath.org/authors/?q=ai:luca.florian"Togbé, Alain"https://zbmath.org/authors/?q=ai:togbe.alainSummary: In this paper, we find all solutions of the exponential Diophantine equation \(B^x_{n+1}-B^x_n = B_m\) in positive integer variables \((m,n,x)\), where \(B_k\) is the \(k\)-th term of the Balancing sequence.A note on the exponential Diophantine equation \((a^n - 1)(b^n - 1) = x^2\).https://zbmath.org/1449.110682021-01-08T12:24:00+00:00"Noubissie, Armand"https://zbmath.org/authors/?q=ai:noubissie.armand"Togbé, Alain"https://zbmath.org/authors/?q=ai:togbe.alainSummary: Let \(a\) and \(b\) be two distinct fixed positive integers such that \(\min\{a,b\} > 1\). We show that the equation in the title with \(b \equiv 3 \pmod{12}\) and \(a\) even has no solution in positive integers \((n,x)\). This generalizes a result of it \textit{L. Szalay} [Publ. Math. 57, No. 1--2, 1--9 (2000; Zbl 0961.11013)]. Moreover, we show that this equation in the title with (\(a \equiv 4 \pmod{10}\) and \(b \equiv 0\pmod 5\)) has no solution in positive integer \((n,x)\). We give a necessary and sufficient condition for Diophantine equation \((a^n - 1)(b^n - 1) = x^2\) with \((a \equiv 4\pmod 5\) and \(b \equiv 0\pmod 5)\) or \((a \equiv 3 \pmod 4\) and \(b \equiv 0\pmod 2\)) to have positive integer solutions. Finally, we prove that the equation with \(a\) even, \(\vartheta_2 (b - 1) = 1\) and \(5 \mid b\) has no solution in positive integer \((n,x)\), where \(\vartheta_2\) is the 2-adic valuation.On the \(X\)-coordinates of Pell equations which are rep-digits. II.https://zbmath.org/1449.110542021-01-08T12:24:00+00:00"Luca, Florian"https://zbmath.org/authors/?q=ai:luca.florian"Togan, Sossa Victorin"https://zbmath.org/authors/?q=ai:togan.sossa-victorin"Togbé, Alain"https://zbmath.org/authors/?q=ai:togbe.alainSummary: For a positive integer \(d\) which is not a square, we show that there is at most one value of the positive integer \(X\) participating in the Pell equation \(X^2-dY^2= \pm 4\) which is a rep-digit, that is all its base 10 digits are equal, except for \(d = 2,5,13\).
For Part I, see Publ. Math. 88, No. 3-4, 381--399 (2016; Zbl 1389.11076).A remark on large gaps between primes in arithmetic progressions.https://zbmath.org/1449.110922021-01-08T12:24:00+00:00"Kaptan, Deniz Ali"https://zbmath.org/authors/?q=ai:kaptan.deniz-aliFor relatively prime positive integers \(q\) and \(a\), let \(p'_n\) be the \(n\)th prime congruent to \(a\) modulo \(q\) and put \[ G(x;q,a)=\max_{p'_n\le x}(p'_{n+1}-p'_n). \] The note under review is a record of an observation that the recent work \textit{J. Li} et al. [Q. J. Math. 68, No. 3, 729--758 (2017; Zbl 1426.11091)] on the least prime in an arithmetic progression, directly yields an improvement on the current lower bound for \(G\) due to \textit{A. Zaccagnini} [J. Number Theory 42, No. 1, 100--102 (1992; Zbl 0770.11042)] as follows.
Given \(\epsilon>0\), there exists \(q_0(\epsilon)\) such that for all integers \(q>q_0(\epsilon)\) with no more than \(\exp((\frac12-\epsilon)\log_2q\log_4q/\log_3q)\) distinct prime factors, there holds \[ G(qx;q,a)\gg\varphi(q)\frac{\log x\log_2x\log_4x}{\log_3x}. \]
Reviewer: Gennady Bachman (Las Vegas)Pillai's problem with the Fibonacci and Padovan sequences.https://zbmath.org/1449.110292021-01-08T12:24:00+00:00"Garcia Lomelia, Ana Cecilia"https://zbmath.org/authors/?q=ai:garcia-lomelia.ana-cecilia"Hernández Hernández, Santos"https://zbmath.org/authors/?q=ai:hernandez.santos-hernandez"Luca, Floorian"https://zbmath.org/authors/?q=ai:luca.floorianSummary: Let \((F_m)_{m\geq 0}\) and \((P_n)_{n\geq 0}\) be the Fibonacci and Padovan sequences given by the initial conditions \(F_0 = 0,F_1 = 1,P_0 = 0,P_1 = P_2 = 1\) and the recurrence formulas \(F_{m+2} = F_{m+1} + F_{m} ,P_{n+3} = P_{n+1} + P_n\) for all \(m,n \geq 0\), respectively. In this note we study and completely solve the Diophantine equation \(P_ n - F_m = P_{n_1} - F_{m_1}\) in non-negative integers \((n,m,n_1 ,m_1)\) with \((n,m) \ne (n_1 ,m_1)\).Discoveries on Collatz conjecture.https://zbmath.org/1449.110442021-01-08T12:24:00+00:00"Chen, Tieling"https://zbmath.org/authors/?q=ai:chen.tielingSummary: The paper provides two new points of view on the problem of \(3n+1\). One is a place shift problem demonstrating that the operations involved in the problem of \(3n+1\) and its generalizations are equivalent to shifting sequences of digits expressed in a number system with a proper base. The other one is a problem about the existence of a particular hypothetical graph for the number theoretical function of the problem of \(3n+1\). Some new discoveries about the problem and its generalizations are presented in the paper. These include the phenomenon that the majority of the numbers entering the final cycle through the number 5 on the classical problem of \(3n+1\) and the phenomenon that the majority of the numbers entering a specific cycle on a generalized problem in the number system with base 3.Fibonacci numbers which are products of two balancing numbers.https://zbmath.org/1449.110282021-01-08T12:24:00+00:00"Erduvan, Fatih"https://zbmath.org/authors/?q=ai:erduvan.fatih"Keskin, Refik"https://zbmath.org/authors/?q=ai:keskin.refikSummary: The Fibonacci sequence \((F_n)\) is defined by \(F_0 = 0, F_1 = 1\) and \(F_n =F_{n-1} + F_{n-2}\) for \(n\geq 2\). The balancing number sequence \((B_n)\) is defined by \(B_0 = 0,B_1 = 1\) and \(B_n=6B_{n-1}-B_{n-2}\) for \(n\geq 2\). In this paper, we find all Fibonacci numbers which are products of two balancing numbers. Also we found all balancing numbers which are products of two Fibonacci numbers. More generally, taking \(k,m,n\) as positive integers, it is proved that \(F_k = B_m B_n\) implies that \((k,m,n) = (1,1,1),(2,1,1)\) and \(B_k = F_m F_n\) implies that \((k,m,n) = (1,1,1),(1,1,2),(1,2,2),(2,3,4)\).Korselt rational bases of prime powers.https://zbmath.org/1449.111152021-01-08T12:24:00+00:00"Ghanmi, Nejib"https://zbmath.org/authors/?q=ai:ghanmi.nejibKorselt proved in 1899 that a squarefree integer \(n\) is a Carmichael numbers -- that is a pseudoprime for any base -- if and only if \(p-1\mid N-1\) for all \(p\mid N\). Generalizing this properly (but without connections to pseudo-primality), the author considers the case that \(a_2p-a_1\) divides \(a_2n-a_1\) for any \(p\vert n\) and defines such \(n\) as being \(\frac{a_1}{a_2}\)-Korselt. He studies this property for \(n\) being a prime power and shows that infinitely many prime powers are \(\alpha\)-Korselt for any given nonzero rational \(\alpha=\frac{a_1}{a_2}\). The proofs only require elementary number theory.
Reviewer: Alexander Hulpke (Fort Collins)Perfect Pell and Pell-Lucas numbers.https://zbmath.org/1449.110262021-01-08T12:24:00+00:00"Bravo, Jhon J."https://zbmath.org/authors/?q=ai:bravo.jhon-jairo"Luca, Florian"https://zbmath.org/authors/?q=ai:luca.florianThe Pell sequence \(\{P_n\}_{n=0}^{\infty}\) is given by the recurrence \(P_n=2P_{n-1}+P_{n-2}\) with initial conditions \(P_0=0\) and \(P_1=1\), and its associated Pell-Lucas sequence \(\{Q_n\}_{n=0}^{\infty}\) is given by the same recurrence relation but with initial conditions \(Q_0=2\) and \(Q_1=2\). The authors presently continue related wok of theirs on the Fibonacci and Lucas sequences, and they show that 6 is the only perfect number appearing in \(\{P_n\}_{n=0}^{\infty}\) and \(\{Q_n\}_{n=0}^{\infty}\).
Reviewer: Andreas N. Philippou (Patras)On the distribution of consecutive primitive roots.https://zbmath.org/1449.110052021-01-08T12:24:00+00:00"Liu, Huaning"https://zbmath.org/authors/?q=ai:liu.huaning"Jing, Mengyao"https://zbmath.org/authors/?q=ai:jing.mengyaoSummary: Let \(p\) be a prime, \(f (x) \in {\mathcal{F}_p}[x]\) with degree \(D \ge 1\). Let \(k \ge 2\) be an integer, and let \({l_1}, {l_2}, \cdots, {l_k}\) be distinct elements in \({\mathcal{F}_p}\). Suppose that at least one of the following conditions holds: (i) \(f (x)\) is irreducible; (ii) \(f (x)\) has no multiple zeros in \({\mathcal{\bar F}_p}\), \(D < p\) and \(k = 2\); (iii) \(f (x)\) has no multiple zeros in \({\mathcal{\bar F}_p}\), and \( (4k)^D < p\). We prove that for all primes \(p > \max\{e^{2^{3k}}, (kD)^{27}\}\) there exists \(n \in {\mathcal{F}_p}\) such that \(f (n+{l_1}), f (n+{l_2}), \cdots, f (n+{l_k})\) all are primitive roots of modulo \(p\).The adjacency-Jacobsthal sequence in finite groups.https://zbmath.org/1449.110792021-01-08T12:24:00+00:00"Karaduman, Erdal"https://zbmath.org/authors/?q=ai:karaduman.erdal"Aküzüm, Yeşim"https://zbmath.org/authors/?q=ai:akuzum.yesim"Deveci, Ömür"https://zbmath.org/authors/?q=ai:deveci.omurSummary: The adjacency-Jacobsthal sequence and the adjacency-Jacobsthal matrix were defined by \textit{Ö. Deveci} and \textit{G. Artun} (see [Commun. Algebra 47, No. 11, 4520--4532 (2019; Zbl 07098140)]). In this work, we consider the cyclic groups which are generated by the multiplicative orders of the adjacency-Jacobsthal matrix when read modulo \(\alpha\) (\(\alpha > 1\)). Also, we study the adjacency-Jacobsthal sequence modulo \(\alpha\) and then we obtain the relationship among the periods of the adjacency-Jacobsthal sequence modulo \(\alpha\) and the orders of the cyclic groups obtained. Furthermore, we redefine the adjacency-Jacobsthal sequence by means of the elements of 2-generator groups which is called the adjacency-Jacobsthal orbit. Then we examine the adjacency-Jacobsthal orbit of the finite groups in detail. Finally, we obtain the periods of the adjacency-Jacobsthal orbit of the dihedral group \(D_{10}\) as applications of the results obtained.Phase transitions on \(C^*\)-algebras arising from number fields and the generalized Furstenberg conjecture.https://zbmath.org/1449.460572021-01-08T12:24:00+00:00"Laca, Marcelo"https://zbmath.org/authors/?q=ai:laca.marcelo"Warren, Jacqueline M."https://zbmath.org/authors/?q=ai:warren.jacqueline-mFor an algebraic number field \(K\) with ring of integers \(O_K\), the (multiplicative) monoid \(O_K^{\times}\) of non-zero integers action on the (additive) group \(O_K\) gives rise to the semi-direct product \(O_K\rtimes O_K^{\times}\), called here the ``affine'' or ``\(ax+b\)'' monoid of algebraic integers in \(K\). The Toeplitz-like \(C^*\)-algebra generated by the left regular representation of the \(ax+b\) monoid acting by isometries on \(\ell^2(O_K\rtimes O_K^{\times})\) was studied by \textit{J. Cuntz} et al. [Math. Ann. 355, No. 4, 1383--1423 (2013; Zbl 1273.22008)], who analysed the equilibrium states of the time evolution on this \(C^*\)-algebra determined by the absolute norm, and characterized the simplex of KMS equilibrium states of this dynamical system for any inverse temperature \(\beta\in(0,\infty]\).
In the paper under review, the low-temperature range of the classification of KMS equilibrium states is studied, using the parametrization in terms of tracial states of direct sums of group \(C^*\)-algebras. Because of the action of units arising here, a higher-dimensional version of Furstenberg's seminal conjecture on rigidity for probability measures on the circle invariant under the multiplicative action of a non-lacunary semigroup of integers [\textit{H. Furstenberg}, Math. Syst. Theory 1, 1--49 (1967; Zbl 0146.28502)] enters the picture. The main results classify the behaviours arising in terms of the ideal class group, the degree, and the unit rank of \(K\), and an explicit description of the primitive ideal space of the associated transformation group \(C^*\)-algebra for number fields of unit rank at least \(2\) that are not complex multiplication fields.
Reviewer: Thomas B. Ward (Leeds)Power integral bases in cubic and quartic extensions of real quadratic fields.https://zbmath.org/1449.111042021-01-08T12:24:00+00:00"Gaál, István"https://zbmath.org/authors/?q=ai:gaal.istvan"Remete, László"https://zbmath.org/authors/?q=ai:remete.laszloSummary: Investigations of monogenity and power integral bases were recently extended from the absolute case (over \(\mathbb{Q}\)) to the relative case (over algebraic number fields). Formerly, in the relative case we only succeeded in calculating generators of power integral bases when the ground field is an imaginary quadratic field. This is the first case when we consider monogenity in the more difficult case, in extensions of real quadratic fields. We give efficient algorithms for calculating generators of power integral bases in cubic and quartic extensions of real quadratic fields, more exactly in composites of cubic and quartic fields with real quadratic fields. In case the quartic field is totally complex, we present an especially simple algorithm. We illustrate our method with two detailed examples.Solutions of an indeterminate equation involving the Euler function.https://zbmath.org/1449.110122021-01-08T12:24:00+00:00"Yang, Hai"https://zbmath.org/authors/?q=ai:yang.hai"Li, Jiao"https://zbmath.org/authors/?q=ai:li.jiao"Gao, Xiaomei"https://zbmath.org/authors/?q=ai:gao.xiaomeiSummary: Let \(\varphi (n)\) be Euler function, the main purpose of this paper is to study the solvability of the equation \(\varphi (xyz) = 8 (\varphi (x) + \varphi (y) + \varphi (z))\). By using the elementary methods, all of the positive integer solutions satisfying \(x \ge y \ge z\) are obtained.A note on Pósa's inequality.https://zbmath.org/1449.110172021-01-08T12:24:00+00:00"Zhang, Shaohua"https://zbmath.org/authors/?q=ai:zhang.shaohuaSummary: The problem involving prime inequalities is investigated. Several new inequalities about prime numbers are obtained, which improve some previous works.Kernel of a class of linearized polynomials.https://zbmath.org/1449.111082021-01-08T12:24:00+00:00"Jin, Yong"https://zbmath.org/authors/?q=ai:jin.yong"Jiang, Jingjing"https://zbmath.org/authors/?q=ai:jiang.jingjingSummary: Let \(\boldsymbol{C}\) be the companion matrix of a linearized polynomial and \(\boldsymbol{I}\) be the identity matrix of the same order. We gave a necessary and sufficient condition for its reversibility by analyzing the structure of \(\boldsymbol{C} + \boldsymbol{I}\). Based on this, we gave linearized polynomials whose kernel had trivial intersection with the kernel of trace function.Diophantine inequality with mixed powers \(1,1,3\) and \(k\).https://zbmath.org/1449.110712021-01-08T12:24:00+00:00"Gao, Fang"https://zbmath.org/authors/?q=ai:gao.fangSummary: Let \(k\) be an positive integer with \(k \ge 3\) and \(\eta\) be any real number. Supposing that \({\lambda_1}, {\lambda_2}, {\lambda_3}, {\lambda_4}\) are non-zero real numbers, not all of them have the same sign and \({\lambda_1}/{\lambda_2}\) is irrational. It is proved that the inequality \(|{\lambda_1}{p_1} + {\lambda_2}{p_2} + {\lambda_3}{p_3^3} + {\lambda_4}{p_4^k} + \eta| < (\max {p_j})^{-\sigma}\) has infinitely many solutions in prime variables \({p_1}, {p_2}, {p_3}, {p_4}\), where \(\sigma = \frac{1}{16}\left (\frac{k + 4}{k}\right) + \varepsilon\), \(\varepsilon > 0\).On the trace of products of \(k\) elements in finite fields.https://zbmath.org/1449.110942021-01-08T12:24:00+00:00"Li, Keyao"https://zbmath.org/authors/?q=ai:li.keyao"Liu, Hua'ning"https://zbmath.org/authors/?q=ai:liu.huaningSummary: Let \(p\) be an odd prime, \(r \ge 2\) be an integer, \(q = {p^r}\), \({\mathbb{F}_q}\) be a finite field and let \(\mathrm{Tr}\) be the trace function from \({\mathbb{F}_q}\) to \({\mathbb{F}_p}\). Let \(k \ge 2\) be an integer, and let \({C_1}, {C_2},\cdots,{C_k}\) be nonempty subsets of \(\mathbb{F}_q^*\). This paper studied the distribution of \({\mathrm{Tr}} ({c_1}{c_2}\cdots{c_k})\), where \({c_1} \in {C_1}, {c_2} \in {C_2},\cdots, {c_k} \in {C_k}\), and showed that \({\mathrm{Tr}} ({c_1}{c_2}\cdots{c_k})\) is well-distributed in \({\mathbb{F}_p}\) under certain conditions.On the nonlinear exponential sums involving the Liouville function.https://zbmath.org/1449.110812021-01-08T12:24:00+00:00"Huang, Jing"https://zbmath.org/authors/?q=ai:huang.jing"Yan, Xiaofei"https://zbmath.org/authors/?q=ai:yan.xiaofei"Zhang, Deyu"https://zbmath.org/authors/?q=ai:zhang.deyuSummary: Let \(\lambda (n)\) be the Liouville function. The main purpose of this paper is to consider the case that \(\beta\) is variable. The main techniques we used are Vaughan's identity and Perron's formula. We prove a nontrivial upper bound for the nonlinear exponential sum of the Liouville function.A new view on fuzzy codes and its application.https://zbmath.org/1449.110762021-01-08T12:24:00+00:00"Amudhambigai, B."https://zbmath.org/authors/?q=ai:amudhambigai.b"Neeraja, A."https://zbmath.org/authors/?q=ai:neeraja.aSummary: In this paper, the notion of fuzzy complement, fuzzy intersection and
fuzzy union on fuzzy codes are studied with their respective axioms and also the
arithmetic operations on fuzzy codes are given. The role of these operators on the
dual of fuzzy codes are studied and finally the concept of super increasing sequence
of fuzzy codes is introduced along with its application.On constructing two classes of permutation polynomials over finite fields.https://zbmath.org/1449.111062021-01-08T12:24:00+00:00"Cheng, Kaimin"https://zbmath.org/authors/?q=ai:cheng.kaiminSummary: In this paper, we construct two classes of permutation polynomials over finite fields. First, by one well-known lemma proposed by previous researchers, we characterize one class of permutation polynomials of the finite field. Second, by using the onto property of functions related to the elementary symmetric polynomial in multivariable and the general trace function, we construct another class of permutation polynomials of the finite field. This extends the results of previous researchers to the more general cases.Constructions of Sidon spaces and cyclic subspace codes.https://zbmath.org/1449.111112021-01-08T12:24:00+00:00"Zhang, He"https://zbmath.org/authors/?q=ai:zhang.he"Cao, Xiwang"https://zbmath.org/authors/?q=ai:cao.xiwangSummary: In this paper, we present a construction of Sidon spaces and Sidon sets. By using this Sidon spaces, we give some new cyclic subspace codes with size \(\tau \cdot \frac{{q^n}-1}{q-1}\) and minimum distance \(2k-2\), where \(\tau\) is a positive integer. Furthermore, we construct the cyclic subspace code with size \(2\tau \cdot \frac{{q^n}-1}{q-1}\) and with the minimum distance \(2k-2\).Maximum gaps in a class of ternary cyclotomic polynomials.https://zbmath.org/1449.110472021-01-08T12:24:00+00:00"Zhang, Bin"https://zbmath.org/authors/?q=ai:zhang.bin.1|zhang.bin.3|zhang.bin.2|zhang.bin.4Summary: Let the maximum gap of the \(n\)-th cyclotomic polynomial \({\Phi_n} (x)\), written as \(g (\Phi_n)\), be the maximum of the differences (gaps) between two consecutive exponents occurring in \({\Phi_n} (x)\). In this paper, we provide an exact expression for \(g (\Phi_{3 \cdot 5 \cdot r})\) and determine the number of maximum gaps in \(g (\Phi_{3 \cdot 5 \cdot r}) (x)\), where \(r \ge 7\) is a prime. And we present a conjecture for the maximum gaps and the number of maximum gaps of cyclotomic polynomials with any order under a certain condition.Gowers norm and pseudorandom measures of subsets.https://zbmath.org/1449.110802021-01-08T12:24:00+00:00"Liu, Huaning"https://zbmath.org/authors/?q=ai:liu.huaning"Qi, Yuchan"https://zbmath.org/authors/?q=ai:qi.yuchanSummary: Let \(A \subset {\mathbb{Z}_N}\), and \[{f_A} (s) = \begin{cases}1 - {\frac{|A|}{N}}, {\mathrm{for}}\;s \in A, \\ -{\frac{|A|}{N}}, {\mathrm{for}}\; s \not\in A.\end{cases}\] We define the pseudorandom measure of order \(k\) of the subset \(A\) as follows, \[{P_k} (A, N) = \max\limits_D \left|\sum\limits_{n \in {\mathbb{Z}_N}} {f_A} (n + {c_1}){f_A} (n + {c_2}) \cdots {f_A} (n + {c_k})\right|,\] where the maximum is taken over all \(D = ({c_1},{c_2},\cdots, {c_k}) \in {\mathbb{Z}^k}\) with \(0 \le {c_1} < {c_2} < \cdots < {c_k} \le N-1\). The subset \(A \subset {\mathbb{Z}_N}\) is considered as a pseudorandom subset of degree \(k\) if \({P_k} (A, N)\) is ``small'' in terms of \(N\). We establish a link between the Gowers norm and our pseudorandom measure, and show that ``good'' pseudorandom subsets must have ``small'' Gowers norm. We give an example to suggest that subsets with ``small'' Gowers norm may have large pseudorandom measure. Finally we prove that pseudorandom subset of degree \(L (k)\) contains an arithmetic progression of length \(k\), where \[L (k) = 2 \cdot {\mathrm{lcm}}\left (2, 4,\cdots, 2\left\lfloor {\frac{k}{2}} \right\rfloor\right)\; {\mathrm{for}}\; k \ge 4,\] and 1cm\( ({a_1}, {a_2}, \cdots {a_l})\) denotes the least common multiple of \({a_1}, {a_2}, \cdots, {a_l}\).A 2-dimensional analogue of Sárközy's theorem in function fields.https://zbmath.org/1449.111012021-01-08T12:24:00+00:00"Li, Guoquan"https://zbmath.org/authors/?q=ai:li.guoquan"Liu, Baoqing"https://zbmath.org/authors/?q=ai:liu.baoqing"Qian, Kun"https://zbmath.org/authors/?q=ai:qian.kun"Xu, Guiqiao"https://zbmath.org/authors/?q=ai:xu.guiqiaoSummary: Let \({\mathbb{F}_q}[t]\) be the polynomial ring over the finite field \({\mathbb{F}_q}\) of \(q\) elements. For \(N \in \mathbb{N}\), let \({\mathbb{G}_N}\) be the set of all polynomials in \({\mathbb{F}_q}[t]\) of degree less than \(N\). Suppose that the characteristic of \({\mathbb{F}_q}\) is greater than 2 and \(A \subseteq \mathbb{G}_N^2\). If \( (d, {d^2}) \notin A - A = \{a - a' : a, a' \in A\}\) for any \(d \in {\mathbb{F}_q}[t]\backslash \{0\}\), we prove that \(|A| \le {C_q^{2N}}\frac{\log N}{N}\), where the constant \(C\) depends only on \(q\). By using this estimate, we extend Sárközy's theorem in function fields to the case of a finite family of polynomials of degree less than 3.A remark on the coefficients of cyclotomic polynomials.https://zbmath.org/1449.110482021-01-08T12:24:00+00:00"Zhang, Bin"https://zbmath.org/authors/?q=ai:zhang.bin.3|zhang.bin.2|zhang.bin.4|zhang.bin.1Summary: Let \({\Phi_n} (x) = \sum\nolimits_i {a (n, i){x^i}} \) be the \(n\)-th cyclotomic polynomial. It is well-known that the first coefficient of \({\Phi_n} (x)\) not in \(\{-1,0,1\}\) is \(a (3 \cdot 5 \cdot 7, 7) = -2\). Let \(p < q < r\) be odd primes. In this note, we give an explicit formula for \(a (pqr, r)\) and prove that for each pair of primes \(p < q\), there exist infinitely many primes \(r\) such that \(a (pqr, r) = -2\).A continued fraction approximation of the Gamma function related to the Gosper's formula.https://zbmath.org/1449.110772021-01-08T12:24:00+00:00"Tian, Dan"https://zbmath.org/authors/?q=ai:tian.dan"Wang, Liantang"https://zbmath.org/authors/?q=ai:wang.liantangSummary: Firstly, a continued fraction approximation of the Gamma function related to the Gosper formula is established, and the best constant and two-sided inequalities about Gamma function are obtained. Then, considering its simplest form, monotonicity, convexity and concavity are obtained.On the depth of finite Schneider and Ruban continued fractions in \(\mathbb{F} (x)\).https://zbmath.org/1449.110182021-01-08T12:24:00+00:00"Tongron, Y."https://zbmath.org/authors/?q=ai:tongron.yanapat"Kanasri, N. R."https://zbmath.org/authors/?q=ai:kanasri.narakorn-rompurk"Laohakosol, V."https://zbmath.org/authors/?q=ai:laohakosol.vichianSummary: Let \(\mathbb{F}\) be a field. Every rational function in \(\mathbb{F} (x)\) has unique finite Schneider continued fraction (SCF) and Ruban continued fraction (RCF) in \(\mathbb{F}((1/x))\) and \(\mathbb{F}((x))\), the completions of \(\mathbb{F} (x)\) with respect to the infinite and the \(x\)-adic valuations, respectively. The explicit constructions of these continued fractions are described. Upper bounds on the depth (length) of finite SCF in both \(\mathbb{F}((1/x))\) and \(\mathbb{F}((x))\) and of finite RCF in \(\mathbb{F}((x))\) are established. We also prove that the depth of the \(x\)-adic RCF of rational functions in \(\mathbb{F} (x)\) evaluated at rational points is bounded, if \(\mathbb{F}\) is finite.Some properties of a class of refined Eulerian polynomials.https://zbmath.org/1449.110402021-01-08T12:24:00+00:00"Sun, Yidong"https://zbmath.org/authors/?q=ai:sun.yidong"Zhai, Liting"https://zbmath.org/authors/?q=ai:zhai.litingSummary: Recently, a new kind of refined Eulerian polynomials was defined, namely, \[A_n (p,q)=\sum\limits_{\pi\in \mathfrak{S}_n}p^{{\mathrm odes} (\pi)}q^{{\mathrm edes} (\pi)}\] for \(n\geq 1\), where \(\mathfrak{S}_n\) is the set of all permutations on \(\{1, 2, \cdots, n\}\), odes \( (\pi)\) and edes \( (\pi)\) enumerate the number of descents of permutation \(\pi\) in odd and even positions, respectively. In this paper, we obtain an exponential generating function for \(A_n (p,q)\) and give an explicit formula for \(A_n (p,q)\) in terms of Eulerian polynomials \(A_n (q)\) and \(C (q)\), the generating function for Catalan numbers. In certain cases, we establish a connection between \(A_n (p,q)\) and \(A_n (p,0)\) or \(A_n (0,q)\), and express the coefficients of \(A_n (0,q)\) by Eulerian numbers \(A_{n,k}\). Consequently, this connection discovers a new relation between Euler numbers \(E_n\) and Eulerian numbers \(A_{n,k}\).Pairs of dual wavelet frames on local fields.https://zbmath.org/1449.420522021-01-08T12:24:00+00:00"Bhat, M. Younus"https://zbmath.org/authors/?q=ai:bhat.mohammad-younusThe author introduces the notion of orthogonal wavelet frames on local fields of positive characteristic and presents an algorithm for the construction of a pair of orthogonal wavelet frames based on polyphase matrices formed by the polyphase components of the wavelet masks. He also gives a general construction algorithm for all orthogonal wavelet tight frames on local fields of positive characteristic from a compactly supported scaling function and investigates their properties by means of the Fourier transform. The motivation for this work are the papers by \textit{F. A. Shah} [Acta Univ. Apulensis, Math. Inform. 49, 47--65 (2017; Zbl 1413.42060)] on orthogonal wavelet frames generated by Walsh polynomials, and \textit{F. A. Shah} and \textit{L. Debnath} [Analysis, München 33, No. 3, 293--307 (2013; Zbl 1277.42047)] on tight wavelet frames on local fields.
Reviewer: Richard A. Zalik (Auburn)Multivariable Hurwitz-Lerch Zeta function and related Apostol-Euler polynomials.https://zbmath.org/1449.110852021-01-08T12:24:00+00:00"Bin-Saad, M. G."https://zbmath.org/authors/?q=ai:binsaad.maged-g|bin-saad.maged-gumman|bin-saad.maged-gumaan"Bin-Alhag, A. Z."https://zbmath.org/authors/?q=ai:bin-alhag.ali-zSummary: The main object of this work is to introduce a new multivariable extension of the Hurwitz-Lerch Zeta function. We then systematically investigate its mathematical properties and give its explicit relationship with new defined Apostol-Euler polynomials of several variables. We also consider some important special cases.On the applications of the linear recurrence relationships to pseudoprimes.https://zbmath.org/1449.110222021-01-08T12:24:00+00:00"He, Tianxiao"https://zbmath.org/authors/?q=ai:he.tianxiao"Shiue, Peter J. S."https://zbmath.org/authors/?q=ai:shiue.peter-jau-shyongSummary: We present here some results on the applications of linear recursive sequences of order 2 to the Fermat pseudoprimes, Fibonacci pseudoprimes, and Dickson pseudoprimes.Twisted quadratic moments for Dirichlet \(L\)-functions at \(s = 2\).https://zbmath.org/1449.110862021-01-08T12:24:00+00:00"Louboutin, Stéphane R."https://zbmath.org/authors/?q=ai:louboutin.stephane-rLet \(c, n\) be given positive integers and let \(q>2\) be coprime to \(c\). Let \(X_q\) denote the multiplicative group of order \(\phi(q)\) of the Dirichlet characters modulo \(q\). The author shows how to find explicit formulas for the sums \[M(q,c,n) = \frac{2}{\phi(n)} \sum_{\chi\in X_q, \chi(-1)=(-1)^n} \chi(c)\vert L(n,\chi)\vert ^2,\] by expressing \(M(q,c,n)\) in terms of the cotangent sums \[S_{k,l}(c,d) = \sum_{a=1}^{d-1} \cot^k\bigg(\frac{\pi a}{d}\bigg) \cot^l\bigg(\frac{\pi ac}{d}\bigg).\] For small \(c\) and \(n\), such cotangent sums can be evaluated using the observation that \(\cot(\pi k/n), 1\le k\le n-1\) are the roots of \[\frac{(X+i)^n-(X-i)^n}{2in} = X^{n-1}-\frac{(n-1)(n-2)}{6}X^{n-3} + \cdots \in\mathbb{Q}[X].\] Explicit formulae are given for \(M(q,c,2)\) for \(c=1,2,3,4,6\) and \(M(p,5,2)\) for \(p\) prime. For example, if \(p\equiv 1\bmod 5\) then \[M(p,5,2) = \frac{\pi^4}{2250p^4} (p^4+994p^2+1008p-2003)\] and there are similar expressions for the other residue classes of primes modulo 5.
Reviewer: John H. Loxton (Greenwich)Irreducibility criteria for polynomials over some imaginary quadratic fields.https://zbmath.org/1449.111052021-01-08T12:24:00+00:00"Kanasri, N. R."https://zbmath.org/authors/?q=ai:kanasri.narakorn-rompurk"Singthongla, P."https://zbmath.org/authors/?q=ai:singthongla.patiwat"Laohakosol, V."https://zbmath.org/authors/?q=ai:laohakosol.vichianSummary: Let \(K\) be an imaginary quadratic field whose ring of integers \({O_K}\) is a Euclidean domain. In our earlier work, the following four results are proved. First, for \(\beta \in {O_K}\backslash \{0\}\), each element in \({O_K}\) has a base \(\beta\)-expansion whose digits are bounded by certain constants. Secondly, if \(\pi = {\alpha_n}{\beta^n} + {\alpha_{n-1}}{\beta^{n-1}} + \cdots + {\alpha_1}\beta + {\alpha_0}: = f (\beta)\) is a base \(\beta\)-expansion of a prime in \({O_K}\), and the digits \({\alpha_n}\) and \({\alpha_{n-1}}\) satisfy some natural restrictions, then the polynomial \(f (x)\) is irreducible over \(K\). Thirdly, for Gaussian integers, similar base \(\beta\)-expansion but with digits belonging to a complete residue system modulo \(\beta\) is also valid. Lastly, irreducibility results similar to that in the second result continue to hold for Gaussian integers with base expansion as described in the third result. In this paper, we establish some generalizations of the above irreducibility results by considering \(\omega \pi (\omega \in {O_K})\) instead of \(\pi\).An upper bound for the number of solutions of ternary purely exponential Diophantine equations. II.https://zbmath.org/1449.110652021-01-08T12:24:00+00:00"Hu, Yongzhong"https://zbmath.org/authors/?q=ai:hu.yongzhong"Le, Maohua"https://zbmath.org/authors/?q=ai:le.maohuaLet \(a\), \(b\), \(c\) be fixed pairwise coprime positive integers with \(\min\{a,b,c\} > 1\). The authors prove that if \(\max\{a,b,c\} \geq 10^{62}\), then the Diophantine equation
\[a^x + b^y = c^z, \quad x, y, z \in \mathbb{N}, \]
has at most two positive integer solutions \((x, y, z)\). The method is by analyzing the gap rule for solutions of the above equation along the approach given in their previous paper [J. Number Theory 183, 62--73 (2018; Zbl 1433.11036)].
The authors notice that there exist infinitely many triples \((a, b, c)\) such that the above equation has exactly 2 solutions, hence the above bound for the number of solutions should be the best one (except for the case \((a, b, c) = (3, 5, 2)\)), where we have three solutions).
Reviewer: Andrzej Dąbrowski (Szczecin)A congruence involving the quotients of Euler and its applications. III.https://zbmath.org/1449.110382021-01-08T12:24:00+00:00"Cai, Tianxin"https://zbmath.org/authors/?q=ai:cai.tianxin"Zhong, Hao"https://zbmath.org/authors/?q=ai:zhong.hao"Chern, Shane"https://zbmath.org/authors/?q=ai:chern.shaneSummary: In some previous papers, a series of congruences involving binomial coefficients under perfect moduli were introduced. This article generalizes these congruences to cubic cases leading to many new statements. For example, the congruence
\[ \prod_{d\mid n} \binom{kd - 1}{\lfloor d/e\rfloor}^{\mu(n/d} \bmod n^3 \] for
\(e = 2, 3, 4\) and \(6\), and the following congruence
\[ \prod_{d\mid n} \binom{(kd - 1)/2}{(d - 1)/2}^{\mu(n/d)} \equiv 2^{ - (k - 1)\varphi(n)} \begin{cases} \pmod{n^3},\qquad \text{if } 3\nmid n, \\ \pmod{n^3/3},\quad \text{if } 3\mid n. \end{cases} \]
For Parts I and II, see Acta Arith. 103, No. 4, 313--320 (2002; Zbl 1008.11001); ibid. 130, No. 3, 203--214 (2007; Zbl 1135.11002).Moments of additive statistics with respect to the Ewens sampling formula.https://zbmath.org/1449.110912021-01-08T12:24:00+00:00"Manstavičius, Eugenijus"https://zbmath.org/authors/?q=ai:manstavicius.eugenijus"Stepas, Vytautas"https://zbmath.org/authors/?q=ai:stepas.vytautasSummary: The additive semigroup of vectors with non-negative integer coordinates endowed with the Ewens probability measure plays an important role as a probabilistic space for many statistical models. In the present paper, we obtain upper estimates of the power moments of additive statistics defined on the semigroup. The statistics are sums of dependent random variables; however, our results have the form of the Rosenthal and von Bahr-Esseen inequalities. The arguments perfected in probabilistic number theory are adopted in the proofs.A new sum related to Dedekind sums and its values in some special integers.https://zbmath.org/1449.110832021-01-08T12:24:00+00:00"Wang, Tingting"https://zbmath.org/authors/?q=ai:wang.tingting"Guan, Yaliang"https://zbmath.org/authors/?q=ai:guan.yaliangSummary: The aim of this paper is to use the definition of the regular integers modulo a positive integer \(q\) and the analytic method to study the computational problem of one kind of sums related to Dedekind sums, and give some interesting identities for the sums at some special integer points.Module-\(p\) edge magic graceful labelling on graphical passwords.https://zbmath.org/1449.052302021-01-08T12:24:00+00:00"Zhang, Xiaohui"https://zbmath.org/authors/?q=ai:zhang.xiaohui.2"Sun, Hui"https://zbmath.org/authors/?q=ai:sun.hui"Yao, Bing"https://zbmath.org/authors/?q=ai:yao.bingSummary: We considered that each vertex of the ring \({C_n}\) of a super sun-graph \({G_s} ({C_n}, {a_i})\) is added with a path of length two, and the super sun-graph obtained is the characteristic of module-\(p\) edge magic graceful graph. The results show that the super sun-graph constructed with \(n\) trees and joint \({T_i}\) (\(i \in [1, n]\)) with \({n - 1}\) edges is a module-\(p\) edge magic graceful graph.Solutions on Euler function equation \(\varphi (n) = {2^{\omega (n)}}{3^{\omega (n)}}{5^{\omega (n)}}\).https://zbmath.org/1449.110142021-01-08T12:24:00+00:00"Zhang, Sibao"https://zbmath.org/authors/?q=ai:zhang.sibaoSummary: The solvability of the arithmetic functional equation \(\varphi (n) = {2^{\omega (n)}}{3^{\omega (n)}}{5^{\omega (n)}}\) was studied. Using the basic theory of number theory and the method of classification, the specific positive integer solutions of the equation were obtained for \(\omega (n) = 1, 2, 3\) if the equation has positive integer solutions for \(\omega (n) \ge 4\), the explicit form of positive integer solutions was obtained. Therefore, the problem of positive integer solutions of the equation was solved.Two equations on generalized Euler function \({\varphi_2} (n)\).https://zbmath.org/1449.110152021-01-08T12:24:00+00:00"Zhang, Sibao"https://zbmath.org/authors/?q=ai:zhang.sibao"Yoldax, Akim"https://zbmath.org/authors/?q=ai:yoldax.akimSummary: The solvability of two equations \({\varphi_2} (x-{\varphi_2} (x)) = 2\) and \({\varphi_2} ({\varphi_2} (x-{\varphi_2} (x))) = 2\) on generalized Euler function \({\varphi_2} (n)\) was discussed. The equation \({\varphi_2} (x-{\varphi_2} (x)) = 2\) has 5 solutions and the equation \({\varphi_2} ({\varphi_2} (x-{\varphi_2} (x))) = 2\) has 26 solutions, which were given based on elementary methods.Fibonacci numbers in RNA imply the one number model of the genetic code.https://zbmath.org/1449.920342021-01-08T12:24:00+00:00"Négadi, Tidjani"https://zbmath.org/authors/?q=ai:negadi.tidjaniSummary: In this short paper, we present an interesting and welcome connection between two different approaches to the mathematical structure of the standard genetic code, we have considered in the last years. The first one relies on the use of the unique number 23! and the second is based on the atomic composition of the four ribonucleotides UMP, CMP, AMP and GMP, the building-blocks of RNA, where several Fibonacci numbers are seen to occur.Rational solutions of the Diophantine equations \(f(x)^2\pm f(y)^2=z^2\).https://zbmath.org/1449.110622021-01-08T12:24:00+00:00"Youmbai, Ahmed El Amine"https://zbmath.org/authors/?q=ai:youmbai.ahmed-el-amine"Behloul, Djilali"https://zbmath.org/authors/?q=ai:behloul.djilaliIn the paper under review the authors show that for any \(n\ge 1\), there are polynomials \(f(x)\in {\mathbb{Z}}[x]\) of degree \(n\) without multiple roots such that the title equation has infinitely many non-trivial rational solutions \((x,y,z)\). For the proof, they construct some particular polynomials which for certain evaluations of \(x\) and \(y\) in terms of a third parameter \(T\), the resulting equation in \((z,T)\) yields an elliptic curve with a particular rational point on it which is not torsion; hence, any elliptic curve multiple of it yields a distinct solution to the desired equation. Some of the calculations were carried out with Magma.
Reviewer: Florian Luca (Johannesburg)Proof of a conjecture on a congruence modulo 243 for overpartitions.https://zbmath.org/1449.111032021-01-08T12:24:00+00:00"Huang, Xiaoqian"https://zbmath.org/authors/?q=ai:huang.xiaoqian"Yao, Olivia X. M."https://zbmath.org/authors/?q=ai:yao.olivia-xiang-meiAn overpartition of a positive integer \(n\) is a partition of \(n\) in which the first occurrence of a part may be overlined. Let \(\overline{p}(n)\) denote the number of overpartitions of \(n\). \textit{J. F. Fortin} et al. [Ramanujan J. 10, No. 2, 215--235 (2005; Zbl 1079.05003)], \textit{J.-F. Fortin} et al. [Ramanujan J. 10, 215--235 (2005; Zbl 1079.05003)], \textit{M. D. Hirschhorn} and \textit{J. A. Sellers} [J. Comb. Math. Comb. Comput. 53, 65--73 (2005; Zbl 1086.11048)] established some congruences modulo powers of 2 for \(\overline{p}(n)\). \textit{E. X. W. Xia} and \textit{O. X. M. Yao} [J. Number Theory 133, 1932--1949 (2013; Zbl 1275.11136)] and \textit{E. X. W. Xia} [Ramanujan J. 42, 301--323 (2017; Zbl 1422.11214)] found several congruences modulo powers of 2 and 3. Moreover, \textit{E. X. W. Xia} [ibid.] conjectured that, for \(n \ge 0\), \(\overline{p}(96n+76)\equiv 0 \mod 243.\)
In the paper under review the authors confirm this conjecture by using theta function identities and the \((p, k)\)-parametrization of theta functions due to \textit{A. Alaca, Ş. Alaca,} and \textit{K. S. Williams} [Acta Arith. 124, 177--195 (2006; Zbl 1127.11035)], \textit{Ş. Alaca} and \textit{K. S. Williams} [Funct. Approximatio, Comment. Math. 43, 45--54 (2010; Zbl 1213.11087)].
Reviewer: Mihály Szalay (Budapest)A new Gaussian Fibonacci matrices and its applications.https://zbmath.org/1449.110342021-01-08T12:24:00+00:00"Prasad, B."https://zbmath.org/authors/?q=ai:prasad.baleshwar|prasad.b-s-r-v|prasad.bhikhari|prasad.baji-nath|prasad.b-v-s-s-s|prasad.bandhu|prasad.bhagwati|prasad.b-k-raghu|prasad.biren|prasad.bhagwat|prasad.brij-nandan|prasad.birendra|prasad.b-e|prasad.baij-nath|prasad.b-g|prasad.b-v-n-s|prasad.bhagwan|prasad.b-jaya|prasad.b-d-c-n|prasad.bhanu|prasad.b-v-l-s|prasad.b-a|prasad.b-s-v|prasad.banu|prasad.b-hari|prasad.b-r-guruSummary: In this paper, we introduced a new Gaussian Fibonacci matrix, \(G^n\) whose elements are Gaussian Fibonacci numbers and we developed a new coding and decoding method followed from this Gaussian Fibonacci matrix, \(G^n\). We established the relations between the code matrix elements, error detection and correction for this coding theory. Correction ability of this method is 93.33\%.Representation of the elements of the finite field \(\mathbb{F}_p\) by fractions.https://zbmath.org/1449.110062021-01-08T12:24:00+00:00"Louboutin, S."https://zbmath.org/authors/?q=ai:louboutin.stephane-r"Murchio, A."https://zbmath.org/authors/?q=ai:murchio.aThe authors reword Thue's Lemma, see e.g. \textit{V. Shoup} [A computational introduction to number theory and algebra. 2nd ed. Cambridge: Cambridge University Press (2009; Zbl 1196.11002)] to state that for any odd prime \(p\), \[ (\mathbb{Z}/p \mathbb{Z})^* = \{\pm a/b \mid 1 \le a, b < \sqrt{p}\}.\] However, when \(p\) is replaced by \(n= 2p\) (and \(p>3\)), by considering the element \(p-2\) in \((\mathbb{Z}/p \mathbb{Z})^*\) they show that the bound of \(\sqrt{n}\) must here be replaced by at least the ceiling of \(\dfrac{p+3-\frac{p}{3}}{3}\), where \( \frac{p}{3}\) is the Legendre symbol, and give a conjecture for the exact bound in general.
Reviewer: Thomas A. Schmidt (Corvallis)On the Diophantine equation \(L_n-L_m = 2\cdot 3^a\).https://zbmath.org/1449.110252021-01-08T12:24:00+00:00"Bitim, Bahar Demirtürk Bitim"https://zbmath.org/authors/?q=ai:bitim.bahar-demirturk-bitimLet \(L_{{\kern 1pt} k} \) be the Lucas numbers. The author solved the Diophantine equation \(L_{n} -L_{m} =2\cdot 3^{{\kern 1pt} a} \) for nonnegative integers \(n,m,a;\; \, n>m\). The solutions for \((n,m,a)\) are the triplets \((3,0,0),(2,1,0),(4,1,1),(7,5,2),(8,7,2)\).
Reviewer: Khristo N. Boyadzhiev (Ada)New relation formula for generating functions.https://zbmath.org/1449.130182021-01-08T12:24:00+00:00"Chammam, Wathek"https://zbmath.org/authors/?q=ai:chammam.wathekSummary: In this paper, we develop a new relation between certain types of generating functions using formal algorithmic methods. As an application, we give a relation between the generating function and finite-type relations between polynomial sequences.Primes of the form \(kM^n+n\).https://zbmath.org/1449.110952021-01-08T12:24:00+00:00"Sun, Xue-Gong"https://zbmath.org/authors/?q=ai:sun.xuegongSummary: \textit{P. Erdős} and \textit{A. M. Odlyzko} [J. Number Theory 11, 257--263 (1979; Zbl 0405.10036)] proved that odd integers \(k\) such that \(k2^n+1\) is prime for some positive integer \(n\) have a positive lower density. We prove that for sufficiently large \(x\), the number of integers \(k\leq x\) such that \(k\) is relatively prime to \(M\) and such that \(kM^n+n\) is prime for some positive integer \(n\) is at least \(C(M)x\) for some constant \(C(M)\) depending only on \(M\).On linear independence of trigonometric numbers.https://zbmath.org/1449.110782021-01-08T12:24:00+00:00"Berger, Arno"https://zbmath.org/authors/?q=ai:berger.arnoSummary: A necessary and sufficient condition is established for 1, cos\((\pi r_1)\), and cos\((\pi r_2)\) to be rationally independent, where \(r_1, r_2\) are rational numbers. The elementary computational argument yields linear independence over larger number fields as well.The order of appearance of factorials in the Fibonacci sequence and certain Diophantine equations.https://zbmath.org/1449.110332021-01-08T12:24:00+00:00"Pongsriiam, Prapanpong"https://zbmath.org/authors/?q=ai:pongsriiam.prapanpongSummary: Let \(F_n\) be the \(n\)th Fibonacci number. The order (or the rank) of appearance of \(m\) in the Fibonacci sequence, denoted by \(z(m)\), is the smallest positive integer \(k\) such that \(m|F_k\). In this article, we obtain a complete formula for the \(p\)-adic valuations of \(z(m!)\) for all \(m\in\mathbb{N}\) and apply it to solve some Diophantine equations involving \(z(m!)\).The factorization of the cubic symmetric polynomial \({x^3} + {y^3} + {z^3} - 3xyz\) and its application. II.https://zbmath.org/1449.110502021-01-08T12:24:00+00:00"Liu, Heguo"https://zbmath.org/authors/?q=ai:liu.heguo"Gao, Rui"https://zbmath.org/authors/?q=ai:gao.rui"Xu, Xingzhong"https://zbmath.org/authors/?q=ai:xu.xingzhong"Luo, Xiaoliang"https://zbmath.org/authors/?q=ai:luo.xiaoliangSummary: As a continuation of our previous work, we give more applications of the factorization of the cubic symmetric polynomial \({x^3} + {y^3} + {z^3} - 3xyz\).Diophantine \(S\)-quadruples with two primes which are twin.https://zbmath.org/1449.110672021-01-08T12:24:00+00:00"Luca, F."https://zbmath.org/authors/?q=ai:luca.floorian|luca.florianSummary: We show that there are only finitely many pairs of twin primes \((p, p+2)\) such that there exists an \(S\)-Diophantine quadruple in the sense of \textit{L. Szalay} and \textit{V. Ziegler} [Publ. Math. 83, No. 1--2, 97--121 (2013; Zbl 1274.11095)] for the set \(S\) of integers composed only of primes \(p\) and \(p+2\).On the Diophantine equation \(X^2 + 4Y^4 = pZ^4\).https://zbmath.org/1449.110572021-01-08T12:24:00+00:00"Guan, Xungui"https://zbmath.org/authors/?q=ai:guan.xunguiSummary: Let \(p\) be a prime and \(p = 4A^2 + 1\), \(2\nmid A\), \(A \in \mathbb N^*\). In this paper, by using some results of quadratic and quartic Diophantine equations, we prove that: except \( (X, Y, Z) = (1, A, 1)\), if \(A \equiv 1\pmod 4\), then the equation
\[ G: X^2 + 4Y^4 = pZ^4\quad\text{with }\gcd(X, Y, Z) = 1, \]
has at most positive integer solutions satisfying
\[ X = |p (a^2 - b^2)^2-4 (A (a^2-b^2) \pm ab)^2|, \]
\[ Y^2 = A (a^2 - b^2)^2 \pm 2ab (a^2 - b^2) - 4a^2b^2A; \]
if \(A \equiv 3\pmod 4\), then the equation \(G\) has at most positive integer solutions satisfying
\[ X = |4a^2b^2A - (4abA \pm (a^2 - b^2))^2|, \]
\[Y^2 = 4a^2b^2A \pm 2ab (a^2 - b^2) - A (a^2 - b^2)^2, \]
\[ Z = a^2 + b^2, \]
where \(a, b \in \mathbb N^*\) and \(a > b\), \(\gcd(a, b) = 1\), \(2\nmid (a + b)\). Additionally, all positive integer solutions of the equation \(G\) when \(p = 5\) are given.A new matrix inversion for Bell polynomials and its applications.https://zbmath.org/1449.150062021-01-08T12:24:00+00:00"Wang, Jin"https://zbmath.org/authors/?q=ai:wang.jinSummary: The present paper gives a new Bell matrix inversion which arises from the classical Lagrange inversion formula. Some new relations for the Bell polynomials are obtained, including a Bell matrix inversion in closed form and an inverse form of the classical Faa di Bruno formula.On the values of representation functions.https://zbmath.org/1449.110212021-01-08T12:24:00+00:00"Jiang, Xing Wang"https://zbmath.org/authors/?q=ai:jiang.xing-wangLet \(\mathbb{N}\) be the set of non-negative integers. For a set \(\mathcal{A}\subset \mathbb{N}\), let \(\mathcal{R}_{2}(\mathcal{A}, n)\) and \(\mathcal{R}_{3}(\mathcal{A}, n)\) denote the number of solutions to \(a + a' = n, a, a' \in \mathcal{A}, a < a'\), and \(a \leq a'\), respectively. \textit{G. Dombi} [Acta Arith. 103, No. 2, 137--146 (2002; Zbl 1014.11009)] (resp. \textit{Y. Chen} and \textit{B. Wang} [Acta Arith. 110, No. 3, 299--303 (2003; Zbl 1032.11008)]) answered a problem of Sárközy by showing that there exist two sets \(\mathcal{A}\) and \(\mathcal{B}\) of non-negative integers with infinite symmetric difference such that for all sufficiently large integers \(n\), \(\mathcal{R}_{2}(\mathcal{A}, n) = \mathcal{R}_{2}(\mathcal{B}, n)\) (resp. \(\mathcal{R}_{3}(\mathcal{A}, n) = \mathcal{R}_{3}(\mathcal{B}, n))\). \textit{C. Sándor} [Integers 4, Paper A18, 5 p. (2004; Zbl 1135.11305)] gave precise formulations of these results. Using character, \textit{M. Tang} [Discrete Math. 308, No. 12, 2614--2616 (2008; Zbl 1162.05003)] provided a more natural proof of Sándor's results. \textit{Y. Chen} [Sci. China, Math. 54, No. 7, 1317--1331 (2011; Zbl 1236.11017)] obtained lower bounds of \(\mathcal{R}_{2}(\mathcal{A}, n)\) and \(\mathcal{R}_{3}(\mathcal{A}, n)\). Let \(\mathcal{R}_{\mathcal{A,B}}(n)\) be the number of solutions to \(a+b = n\) with \(a \in \mathcal{A}, b \in\mathcal{B}\). Let \(\mathcal{A}_{0}\) be the set of all non-negative integers \(a\) having even number of ones in their binary representation and \(\mathcal{B}_{0} = \mathbb{N} \smallsetminus \mathcal{A}_{0}\). Chen proposed a question whether there is an absolute constant \(c\) such that for any \(\mathcal{A} \subset \mathbb{N}\) and any positive integer \(N\), if \(\mathcal{R}_{2}(\mathcal{A}, n) = \mathcal{R}_{2}(\mathbb{N} \smallsetminus \mathcal{A}, n)\) for all \(n \geq 2N - 1\) and \(\mathcal{A} \neq \mathcal{A}_{0},\mathcal{B}_{0}\), then \(\mathcal{R}_{2}(\mathcal{A}, n) = \mathcal{R}_{2}(\mathbb{N} \smallsetminus \mathcal{A}, n)\) for all \(n \geq cN \). A similar question was posed for \(\mathcal{R}_{3}(A,n)\). These questions were answered by \textit{Z. Qu} [Discrete Math. 338, No. 4, 571--575 (2015; Zbl 1308.11011)].
In this paper, the lower bounds of results obtained by Chen have been improved and it is shown that if \(\mathcal{A}\) is a subset of \(\mathbb{N}\) and \(N\) is a positive integer such that \(\mathcal{R}_{2}(\mathcal{A}, n) = \mathcal{R}_{2}(\mathbb{N} \smallsetminus \mathcal{A}, n)\) for all \(n \geq 2N - 1\), then for any \(\epsilon > 0\), the set of integers \(n\) with \((\frac{1}{8}-\epsilon) n \leq \mathcal{R}_{2}(\mathcal{A}, n) \leq (\frac{1}{8}+\epsilon)n \) has density one. Further, if \(\mathcal{R}_{3}(\mathcal{A}, n) = \mathcal{R}_{3}(\mathbb{N} \smallsetminus \mathcal{A}, n)\) for all \(n \geq 2N - 1\), applying the same method and adjusting if necessary, one can get the same result.
Reviewer: Ranjeet Sehmi (Chandigarh)On the probabilistic proof of the convergence of the Collatz conjecture.https://zbmath.org/1449.110432021-01-08T12:24:00+00:00"Barghout, Kamal"https://zbmath.org/authors/?q=ai:barghout.kamalSummary: A new approach towards probabilistic proof of the convergence of the Collatz conjecture is described via identifying a sequential correlation of even natural numbers by divisions by 2 that follows a recurrent pattern of the form \(x, 1, x, 1 \ldots\), where \(x\) represents divisions by 2 more than once. The sequence presents a probability of 50:50 of division by 2 more than once as opposed to division by 2 once over the even natural numbers. The sequence also gives the same 50:50 probability of consecutive Collatz even elements when counted for division by 2 more than once as opposed to division by 2 once and a ratio of 3:1. Considering Collatz function producing random numbers and over sufficient number of iterations, this probability distribution produces numbers in descending order that lead to the convergence of the Collatz function to 1, assuming that the only cycle of the function is 1-4-2-1.Expansions of arithmetic functions of several variables with respect to certain modified unitary Ramanujan sums.https://zbmath.org/1449.110102021-01-08T12:24:00+00:00"Tóth, László"https://zbmath.org/authors/?q=ai:toth.laszloSummary: We introduce new analogues of the Ramanujan sums, denoted by \(\widetilde c_q(n)\), associated with unitary divisors, and obtain results concerning the expansions of arithmetic functions of several variables with respect to the sums \(\widetilde c_q(n)\). We apply these results to certain functions associated with \(\sigma^*(n)\) and \(\phi^*(n)\), representing the unitary sigma function and unitary phi function, respectively.