Recent zbMATH articles in MSC 11Ahttps://zbmath.org/atom/cc/11A2021-11-25T18:46:10.358925ZWerkzeugBook review of: B. Hutz, An experimental introduction to number theoryhttps://zbmath.org/1472.000382021-11-25T18:46:10.358925Z"Verhoeff, Tom"https://zbmath.org/authors/?q=ai:verhoeff.tomReview of [Zbl 1443.11001].Crossings and nestings over some Motzkin objects and \(q\)-Motzkin numbershttps://zbmath.org/1472.050092021-11-25T18:46:10.358925Z"Rakoyomamonjy, Paul M."https://zbmath.org/authors/?q=ai:rakoyomamonjy.paul-m"Andriantsoa, Sandrataniaina R."https://zbmath.org/authors/?q=ai:andriantsoa.sandrataniaina-rSummary: We examine the enumeration of certain Motzkin objects according to the numbers of crossings and nestings. With respect to continued fractions, we compute and express the distributions of the statistics of the numbers of crossings and nestings over three sets, namely the set of \(4321\)-avoiding involutions, the set of \(3412\)-avoiding involutions, and the set of \((321,3\bar{1}42)\)-avoiding permutations. To get our results, we exploit the bijection of \textit{P. Biane} [Eur. J. Comb. 14, No. 4, 277--284 (1993; Zbl 0784.05005)] restricted to the sets of \(4321\)- and \(3412\)-avoiding involutions which was characterized by \textit{M. Barnabei} et al. [Adv. Appl. Math. 47, No. 1, 102--115 (2011; Zbl 1225.05242)] and the bijection between \((321,3\bar{1}42)\)-avoiding permutations and Motzkin paths, presented by \textit{W. Y. C. Chen} et al. [J. Comb. 9, No. 2, Research paper R15, 13 p. (2003; Zbl 1023.05002)]. Furthermore, we manipulate the obtained continued fractions to get the recursion formulas for the polynomial distributions of crossings and nestings, and it follows that the results involve two new \(q\)-Motzkin numbers.Number theory. Translated from the Croatian by Petra Švobhttps://zbmath.org/1472.110012021-11-25T18:46:10.358925Z"Dujella, Andrej"https://zbmath.org/authors/?q=ai:dujella.andrejThis book is based on teaching materials from the courses \textit{Number theory} and \textit{Elementary number theory}, which are taught at the undergraduate level studies at the Department of Mathematics, University of Zagreb, and the courses \textit{Diophantine equations} and \textit{Diophantine approximations and applications}, which were taught at the doctoral program of mathematics at that unit. This book is primarily intended for teachers and students of mathematics and related subjects at universities. It can also be useful to advanced high school students who are preparing for mathematics competitions at all levels, from the school level to international competitions, and for doctoral students and scientists in the fields of number theory, algebra and cryptography.
The book is composed by 16 chapters:
1 Introduction; 2 Divisibility; 3 Congruences; 4 Quadratic residues; 5 Quadratic forms; 6 Arithmetical functions; 7 Distribution of primes; 8 Diophantine approximation; 9 Applications of Diophantine approximation to cryptography; 10 Diophantine equations I; 11 Polynomials; 12 Algebraic numbers; 13 Approximation of algebraic numbers; 14 Diophantine equations II; 15 Elliptic curves; 16 Diophantine problems and elliptic curves.
Each chapter is partitioned into several parts and is concluded with a collection of various exercises.
This book is a beautiful invitation to number theory. It provides interesting connections between various fields of number theory. Proofs are presented in a concise form. I think that this is a useful opus for a wide branch of readership interested in number theory.Number systems. A path into rigorous mathematicshttps://zbmath.org/1472.110022021-11-25T18:46:10.358925Z"Kay, Anthony"https://zbmath.org/authors/?q=ai:kay.anthonyPublisher's description: This text aims to introduce number systems to an undergraduate audience in a way that emphasises the importance of rigour, and with a focus on providing detailed but accessible explanations of theorems and their proofs. The book continually seeks to build upon students' intuitive ideas of how numbers and arithmetic work, and to guide them towards the means to embed this natural understanding into a more structured framework of understanding.
The author's motivation for writing this book is that most previous texts, which have complete coverage of the subject, have not provided the level of explanation needed for first-year students. On the other hand, those that do give good explanations tend to focus broadly on Foundations or Analysis and provide incomplete coverage of Number Systems.
Features:
\begin{itemize}
\item Approachable for students who have not yet studied mathematics beyond school
\item Does not merely present definitions, theorems and proofs, but also motivates them in terms of intuitive knowledge and discusses methods of proof
\item Draws attention to connections with other areas of mathematics
\item Plenty of exercises for students, both straightforward problems and more in-depth investigations
\item Introduces many concepts that are required in more advanced topics in mathematics.
\end{itemize}Digital images unveil geometric structures in pairs of relatively prime numbershttps://zbmath.org/1472.110062021-11-25T18:46:10.358925Z"Itzá-Ortiz, Benjamín A."https://zbmath.org/authors/?q=ai:itza-ortiz.benjamin-a"López-Hernández, Roberto"https://zbmath.org/authors/?q=ai:lopez-hernandez.roberto"Miramontes, Pedro"https://zbmath.org/authors/?q=ai:miramontes.pedroFrom the text: The interaction of geometry with other branches of mathematics has proven to be fruitful and mutually beneficial. In this paper, we explore an example of the amazing relationship between number theory and geometry.
We show that in plotting a certain large collection of relatively prime numbers, a family of quadratic arcs emerges. The appearance of these arcs was unexpected, and to the best of our knowledge, has not been observed before. We divide this work into two sections. In the first, we introduce the Bézout transformations and use them to generate a special family of relatively prime numbers. In the second section, we state our main results, which justify the appearance of quadratic arcs in the graphs we have introduced and also their symmetries.A short and easy proof of Morley's congruence theoremhttps://zbmath.org/1472.110072021-11-25T18:46:10.358925Z"Aebi, Christian"https://zbmath.org/authors/?q=ai:aebi.christianSummary: Morley's theorem (1899): The three points of intersection of adjacent trisectors of the angles of any triangle form an equilateral triangle.Addendum and corrigendum to: ``Densities of primes and primitive roots''https://zbmath.org/1472.110082021-11-25T18:46:10.358925Z"Carella, N. A."https://zbmath.org/authors/?q=ai:carella.nelson-aSummary: Let \(u\ne\pm 1\), \(v^2\) be a fixed integer, let \(p\ge 2\) be a prime, and let \(\mathrm{ord}_p(u)=d\mid p-1\) be the order of \(u\bmod p\). This paper provides an effective lower bound \(\#\{p\le x:\mathrm{ord}_p(u)=p-1\}\gg x(\log x)^{-1}\) for the number of primes \(p\le x\) with a fixed primitive root \(u\bmod p\) for all large numbers \(x\ge 1\). The current results in the literature have the lower bound \(\#\{p\le x:\mathrm{ord}_p(u)=p-1\}\gg x(\log x)^{-2}\), and restrictions on the fixed primitive root to a subset of at least three or more integers.
For the original paper, see [the author, ibid. 11, No. 2, 89--108 (2016; Zbl 1395.11008)].The congruence equation \(\overline{a} + \overline{b} \equiv \overline{c} \pmod{p}\)https://zbmath.org/1472.110092021-11-25T18:46:10.358925Z"Chan, Tsz Ho"https://zbmath.org/authors/?q=ai:chan.tsz-hoProof of a supercongruence conjecture of Hehttps://zbmath.org/1472.110102021-11-25T18:46:10.358925Z"Chetry, Arjun Singh"https://zbmath.org/authors/?q=ai:chetry.arjun-singh"Jana, Arijit"https://zbmath.org/authors/?q=ai:jana.arijit"Kalita, Gautam"https://zbmath.org/authors/?q=ai:kalita.gautamThe authors prove the following theorem:
Theorem 1.2 Let \(p\) be an odd prime such that \(p\equiv 5 \pmod 6\). If \(r\) is a positive integer,
then
\[
\sum_{k=0}^{tp^r-1} \left(\frac{(\frac{1}{3})_k}{k!}\right)^3 \equiv \begin{cases}\quad p^r\pmod{p^{\min(6,r+3)}},\quad \text{if }r \text{ is even}, \\ -\frac{p^{r+1}}{3} \Gamma_p(\frac{1}{3})^6 \pmod{p^3},\quad \text{if }r \text{ is odd}, \end{cases}
\]
where \(t=1\) if \(r\) is even, and \(t=2\) if \(r\) is odd.
Using \(\binom{z}{k} = (-1) \frac{(-z)_k}{(1)_k}\) they obtain a recent conjectural supercongruence of \textit{Bing He} [Result. Math. 71, No. 3-4, 1223--1234 (2017; Zbl 1421.11005), Conjecture1.3]
They use the general method introduced by \textit{L. Long} [Pac. J. Math. 249, No. 2, 405--418 (2011; Zbl 1215.33002)] to prove their results. Furthermore, a suitable hypergeometric series identity and properties of gamma function together with recurrence relations of certain rising factorials are used to deduce the main result.On a general Van Hamme-type supercongruencehttps://zbmath.org/1472.110112021-11-25T18:46:10.358925Z"Chetry, Arjun Singh"https://zbmath.org/authors/?q=ai:chetry.arjun-singh"Kalita, Gautam"https://zbmath.org/authors/?q=ai:kalita.gautamIn this paper, we deduce a congruence recurrence relation for truncated hypergeometric series. As consequences, we confirm certain particular cases of a recent conjectural supercongruence of Swisher.
The congruence recurrence relation in question is the following one:
Theorem 1.1. Let \(p\ge 11\) be a a prime with \(p\equiv 2\pmod 3\) and \(r>1\) an integer.
If \(\displaystyle S(n):= \sum_{k=0}^n (6k+1)\frac{(\frac13)_k^6}{k!^6}\), then
\[
S\left(\frac{tp^r - 1}{3}\right) \equiv (-p)^5 S\left(\frac{tp^{r - 2} - 1}{3}\right) \pmod{p^{r+5}},
\]
where \(t=1\), if \(r\) is even, and \(t=2\), if \(r\) is odd.
From this the conjectural supercongruences of \textit{H. Swisher} [Res. Math. Sci. 2, Paper No. 18, 21 p. (2015; Zbl 1337.33005)] and \textit{Bing He} [Result. Math. 71, No. 3-4, 1223--1234 (2017; Zbl 1421.11005)] can be deduced.On generalizing a corollary of Fermat's little theoremhttps://zbmath.org/1472.110122021-11-25T18:46:10.358925Z"Effinger, Gove"https://zbmath.org/authors/?q=ai:effinger.gove-wFermat's little theorem states that, for every prime \(p\nmid a\), it holds that \(a^{p-1}\equiv 1\pmod{p}\). This implies that \(a^p\equiv p\pmod{p}\) for every prime \(p\) and integer \(a\), even if \(p\mid a\).
Fermat's little theorem can be naturally extended to the so-called Euler's theorem which states that, for every prime \(n\) coprime to \(a\), it holds that \(a^{\phi(n)}\equiv 1 \pmod{n}\), where \(\phi\) is the well-know Euler's totient function.
Unlike in the case of Fermat's theorem, Euler's theorem does not imply that \(a^{\phi(n)+1}\equiv n \pmod{n}\) for every pair of integers \(a\) and \(n\). This paper is devoted to investigate this situation.
The author reaches the following result. Given \(n\) and \(n>1\) be integers, let us denote by \(n_a\) the largest factor of \(n\) that contains all the primes in the factorization of \(a\), and those primes only. Then, \(a^{\phi(n)+1}\equiv a\pmod{n}\) if and only if \(n_a\mid a\).
As a corollary, the following result if obtained. Given an integer \(n>1\), there exists \(k>1\) such that \(a^k\equiv a\pmod{n}\) for every \(a\) if and only if \(n\) is square-free. Moreover, \(k=\phi(n)+1\).Proof of two conjectures on supercongruences involving central binomial coefficientshttps://zbmath.org/1472.110132021-11-25T18:46:10.358925Z"Gu, Cheng-Yang"https://zbmath.org/authors/?q=ai:gu.chengyang"Guo, Victor J. W."https://zbmath.org/authors/?q=ai:guo.victor-j-wSummary: In this note we use some \(q\)-congruences proved by the method of `creative microscoping' to prove two conjectures on supercongruences involving central binomial coefficients. In particular, we confirm the \(m=5\) case of Conjecture 1.1 of the second author [Integral Transforms Spec. Funct. 28, No. 12, 888--899 (2017; Zbl 1379.33013)].Some \(q\)-congruences on double basic hypergeometric sumshttps://zbmath.org/1472.110142021-11-25T18:46:10.358925Z"Guo, Victor J. W."https://zbmath.org/authors/?q=ai:guo.victor-j-w"Lian, Xiuguo"https://zbmath.org/authors/?q=ai:lian.xiuguoSummary: We give three \(q\)-congruences on double basic hypergeometric sums. One of them is a \(q\)-analogue of the following supercongruence: for any prime \(p>3\),
\[
\sum\limits^{(p-1)/2}_{k=0}(4k+1)\frac{\left(\frac{1}{2}\right)^4_k}{k!^4}\sum\limits^k_{j=1}\biggl(\frac{1}{(2j-1)^2}-\frac{1}{4j^2}\biggr)\equiv 0\quad\pmod{p^2}.
\]
Our proof uses \(q\)-analogues of two Ramanujan-type supercongruences of Van Hamme and a \(q\)-analogue of a `divergent' Ramanujan-type supercongruence.On a congruence conjecture of Swisherhttps://zbmath.org/1472.110152021-11-25T18:46:10.358925Z"He, Bing"https://zbmath.org/authors/?q=ai:he.bing|he.bing.1|he.bing.4|he.bing.2|he.bing.3Summary: A congruence on a conjecture of van Hamme is established. This result confirms a particular case of a congruence conjecture of \textit{H. Swisher} [Res. Math. Sci. 2, Paper No. 18, 21 p. (2015; Zbl 1337.33005)].On some conjectural supercongruences for sums involving certain rising factorialshttps://zbmath.org/1472.110162021-11-25T18:46:10.358925Z"Jana, Arijit"https://zbmath.org/authors/?q=ai:jana.arijit"Kalita, Gautam"https://zbmath.org/authors/?q=ai:kalita.gautamSummary: We here deduce some supercongruence results for certain sums involving rising factorials using a method similar to the WZ-method. As particular cases, we confirm certain recent conjectural supercongruence of Guo.Comparing multiplicative orders mod \(p\), as \(p\) varieshttps://zbmath.org/1472.110172021-11-25T18:46:10.358925Z"Just, Matthew"https://zbmath.org/authors/?q=ai:just.matthew"Pollack, Paul"https://zbmath.org/authors/?q=ai:pollack.paulSummary: \textit{A. Schinzel} and \textit{J. Wójcik} [Math. Proc. Camb. Philos. Soc. 112, No. 2, 225--232 (1992; Zbl 0770.11001)] have shown that if \(\alpha, \beta\) are rational numbers not 0 or \(\pm 1\), then \(\mathrm{ord}_p (\alpha)=\mathrm{ord}_p(\beta)\) for infinitely many primes \(p\), where \(\mathrm{ord}_p(\cdot)\) denotes the order in \(\mathbb{F}_p^\times\). We begin by asking: When are there infinitely many primes \(p\) with \(\mathrm{ord}_p(\alpha) > \mathrm{ord}_p(\beta)\)? We write down several families of pairs \(\alpha, \beta\) for which we can prove this to be the case. In particular, we show this happens for ``100\%'' of pairs \(A,2\), as \(A\) runs through the positive integers.
We end on a different note, proving a version of Schinzel and Wójcik's theorem for the integers of an imaginary quadratic field \(K\): If \(\alpha, \beta \in \mathcal{O}_K\) are nonzero and neither is a root of unity, then there are infinitely many maximal ideals \(P\) of \(\mathcal{O}_K\) for which \(\mathrm{ord}_P(\alpha) = \mathrm{ord}_P(\beta)\).The valuation tree for \(n^2+7\)https://zbmath.org/1472.110182021-11-25T18:46:10.358925Z"Kozhushkina, Olena"https://zbmath.org/authors/?q=ai:kozhushkina.olena"Hallare, Maila Brucal"https://zbmath.org/authors/?q=ai:brucal-hallare.maila"Long, Jane"https://zbmath.org/authors/?q=ai:long.jane"Moll, Victor H."https://zbmath.org/authors/?q=ai:moll.victor-hugo"Pedjeu, Jean-Claude"https://zbmath.org/authors/?q=ai:pedjeu.jean-claude"Thompson, Bianca"https://zbmath.org/authors/?q=ai:thompson.bianca"Trulen, Justin"https://zbmath.org/authors/?q=ai:trulen.justinSummary: The 2-adic valuation of an integer \(x\) is the highest power of 2 which divides \(x\). It is denoted by \(\nu_2(x)\). The goal of the present work is to describe the sequence \(\{\nu_2(n^2+a)\}\) for \(1\le a \le 7\). The first six cases are elementary. The last case considered here, namely \(a=7\), presents distinct challenges. It is shown here how to represent this family of valuations in the form of an infinite binary tree, with two symmetric infinite branches.On some \(p\)-adic properties and supercongruences of Delannoy and Schröder numbershttps://zbmath.org/1472.110192021-11-25T18:46:10.358925Z"Lengyel, Tamás"https://zbmath.org/authors/?q=ai:lengyel.tamasSummary: The Delannoy number \(d(n)\) is defined as the number of paths from \((0,0)\) to \((n, n)\) with steps \((1,0)\), \((1,1)\), and \((0,1)\), which is equal to the number of paths from \((0,0)\) to \((2n,0)\) using only steps \((1,1)\), \((2,0)\) and \((1,-1)\). The Schröder number \(s(n)\) counts only those paths that never go below the \(x\)-axis. We discuss some \(p\)-adic properties of the sequences \(\{d(p^n)\}_{n \rightarrow \infty}\), and \(\{d(ap^n+b)\}_{n \rightarrow \infty}\) with \(a \in \mathbb{N}\), \((a, p) = 1,b \in \mathbb{Z}\), and prime \(p\). We also present similar \(p\)-adic properties of the Schröder numbers. We provide several supercongruences for these numbers and their differences. Some conjectures are also proposed.Some \(q\)-supercongruences for truncated forms of squares of basic hypergeometric serieshttps://zbmath.org/1472.110202021-11-25T18:46:10.358925Z"Li, Long"https://zbmath.org/authors/?q=ai:li.longSummary: Guo and Zudilin devised a method called `creative microscoping' to establish many \(q\)-supercongruences for truncated basic hypergeometric series. Recently, El Bachraoui utilized the method of creative microscoping to derive some supercongruences for truncated squares of basic hypergeometric series. In this paper, motivated by their work, we give three more such \(q\)-supercongruences.Supercongruences for sums involving Domb numbershttps://zbmath.org/1472.110212021-11-25T18:46:10.358925Z"Liu, Ji-Cai"https://zbmath.org/authors/?q=ai:liu.jicaiSummary: We prove some supercongruence and divisibility results on sums involving Domb numbers, which confirm four conjectures of Z.-W. Sun and Z.-H. Sun. For instance, by using a transformation formula due to Chan and Zudilin, we show that for any prime \(p\geq 5\),
\[
\sum\limits_{k=0}^{p-1}\frac{3k+1}{(-32)^k}\operatorname{Domb}(k)\equiv (-1)^{\frac{p-1}{2}}p+p^3E_{p-3}\pmod{p^4},
\]
which is regarded as a \(p\)-adic analogue of the interesting formula for \(1/\pi\) due to Rogers:
\[
\sum\limits_{k=0}^\infty\frac{3k+1}{(-32)^k}\operatorname{Domb}(k)=\frac{2}{\pi}.
\]
Here \(\operatorname{Domb}(n)\) and \(E_n\) are the famous Domb numbers and Euler numbers.On the congruence of twice a primehttps://zbmath.org/1472.110222021-11-25T18:46:10.358925Z"Lur'e, B. B."https://zbmath.org/authors/?q=ai:lure.boris-beniaminovich"Poretsky, A. M."https://zbmath.org/authors/?q=ai:poretsky.a-mSummary: It is shown that if \(p\) is a prime equal to 5 modulo 8, then \(2p\) cannot be a congruent number. It is also proved that if \(p\) is a prime equal to 1 modulo 8, then \(2p\) can be congruent only if \(p \equiv 1 \pmod{16}\).Proof of a supercongruence via the Wilf-Zeilberger methodhttps://zbmath.org/1472.110232021-11-25T18:46:10.358925Z"Mao, Guo-Shuai"https://zbmath.org/authors/?q=ai:mao.guo-shuaiSummary: In this paper, we prove a supercongruence via the Wilf-Zeilberger method and symbolic summation algorithms in the setting of difference rings. That is, for any prime \(p>3\),
\[
\sum_{n=0}^{(p-1)/2}\frac{3n+1}{(-8)^n}\binom{2n}{n}^3 \equiv p\biggl(\frac{-1}{p}\biggr)+\frac{p^3}{4}\biggl(\frac{2}{p}\biggr)E_{p-3}\biggl(\frac{1}{4}\biggr)\pmod{p^4},
\]
where \(\biggl(\frac{\cdot}{p}\biggr)\) stands for the Legendre symbol, and \(E_n(x)\) are the Euler polynomials. This confirms a special case of a recent conjecture of Z.-W. Sun.An elementary proof of Weisman's congruence when \(p=2\)https://zbmath.org/1472.110242021-11-25T18:46:10.358925Z"Neuberger, Nick"https://zbmath.org/authors/?q=ai:neuberger.nick"Yildiz, Bahattin"https://zbmath.org/authors/?q=ai:yildiz.bahattinSummary: We give an elementary approach to proving divisibility results for a class of binomial sums that are similar to Fleck's congruence. We use tools that are accessible to undergraduate students and in proving the divisibility results, we obtain additional interesting properties that we highlight in several parts of the paper.Primitive roots for Pjateckii-Šapiro primeshttps://zbmath.org/1472.110252021-11-25T18:46:10.358925Z"Sivaraman, Jyothsnaa"https://zbmath.org/authors/?q=ai:sivaraman.jyothsnaaSummary: For any non-integral positive real number \(c\), any sequence \((\lfloor{n^c}\rfloor)_n\) is called a Pjateckii-Šapiro sequence. Given a real number \(c\) in the interval \(\left(1,\frac{12}{11}\right)\), it is known that the number of primes in this sequence up to \(x\) has an asymptotic formula. We would like to use the techniques of Gupta and Murty to study Artin's problems for such primes. We will prove that even though the set of Pjateckii-Šapiro primes is of density zero for a fixed \(c\), one can show that there exist natural numbers which are primitive roots for infinitely many Pjateckii-Šapiro primes for any fixed \(c\) in the interval \(\left(1,\frac{\sqrt{77}}{7}-\frac{1}{4}\right)\).Supercongruences and binary quadratic formshttps://zbmath.org/1472.110262021-11-25T18:46:10.358925Z"Sun, Zhi-Hong"https://zbmath.org/authors/?q=ai:sun.zhihongSummary: Let \(p > 3\) be a prime, and let \(a,b\) be two rational \(p\)-adic integers. We present general congruences for \(\sum_{k=0}^{p-1}\binom{a}{k}\binom{-1-a}{k}\frac{p}{k+b}\pmod{p^2} \). Let \(\{D_n\}\) be the Domb numbers given by \(D_n=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}{k}\binom{2n-2k}{n-k} \). We also prove that
\[\sum_{n=0}^{p-1}\frac{D_n}{16^n}\equiv \sum_{n=0}^{p-1}\frac{D_n}{4^n} \equiv \begin{cases} 4x^2-2p\pmod{p^2} &\text{if } 3\mid p-1\text{ and so }p=x^2+3y^2,\\
0\pmod{p^2} &\text{if }p\equiv 2\pmod 3,\end{cases}\]
which was conjectured by Z. W. Sun.Proof of two congruences concerning Legendre polynomialshttps://zbmath.org/1472.110272021-11-25T18:46:10.358925Z"Wang, Chen"https://zbmath.org/authors/?q=ai:wang.chen"Xia, Wei"https://zbmath.org/authors/?q=ai:xia.wei.3|xia.wei.2|xia.wei|xia.wei.1Summary: The Legendre polynomials \(P_n(x)\) are defined by
\[
P_n(x)=\sum\limits_{k=0}^n \binom{n+k}{k}\binom{n}{k}\left(\frac{x-1}{2}\right)^k\quad (n=0,1,2,\ldots).
\]
In this paper, we prove two congruences concerning Legendre polynomials. For any prime \(p>3\), by using the symbolic summation package Sigma, we show that
\[
\sum\limits_{k=0}^{p-1}(2k+1)P_k(-5)^3\equiv p-\frac{10}{3}p^2q_p(2)\pmod{p^3},
\]
where \(q_p(2)=(2^{p-1}-1)/p\) is the Fermat quotient. This confirms a conjecture of Z.-W. Sun. Furthermore, we prove the following congruence which was conjectured by V. J. W. Guo
\begin{align*}
& \sum\limits_{k=0}^{p-1}(-1)^k(2k+1)P_k(2x+1)^4\\
\equiv p & \sum\limits_{k=0}^{(p-1)/2}(-1)^k\binom{2k}{k}^2(x^2+x)^k(2x+1)^{2k}\pmod{p^3},
\end{align*}
where \(p\) is an odd prime and \(x\) is an integer.Proof of three divisibilities of Franel numbers and binomial coefficientshttps://zbmath.org/1472.110282021-11-25T18:46:10.358925Z"Zhang, Yong"https://zbmath.org/authors/?q=ai:zhang.yong.8|zhang.yong.5|zhang.yong.2|zhang.yong.1|zhang.yong.9|zhang.yong|zhang.yong.13|zhang.yong.14|zhang.yong.12|zhang.yong.7|zhang.yong.11|zhang.yong.10|zhang.yong.4"Yuan, Peisen"https://zbmath.org/authors/?q=ai:yuan.peisen\((q)\)-supercongruences hit againhttps://zbmath.org/1472.110292021-11-25T18:46:10.358925Z"Zudilin, Wadim"https://zbmath.org/authors/?q=ai:zudilin.wadimSummary: Using an intrinsic \(q\)-hypergeometric strategy, we generalise Dwork-type congruences \(H(p^{s+1})/H(p^s)\equiv H(p^s)/H(p^{s-1}) \pmod{p^3}\) for \(s=1,2,\ldots\) and \(p\) a prime, when \(H(N)\) are truncated hypergeometric sums corresponding to the periods of rigid Calabi-Yau threefolds.The eleventh power residue symbolhttps://zbmath.org/1472.110302021-11-25T18:46:10.358925Z"Joye, Marc"https://zbmath.org/authors/?q=ai:joye.marc"Lapiha, Oleksandra"https://zbmath.org/authors/?q=ai:lapiha.oleksandra"Nguyen, Ky"https://zbmath.org/authors/?q=ai:nguyen.ky-t"Naccache, David"https://zbmath.org/authors/?q=ai:naccache.davidSummary: This paper presents an efficient algorithm for computing 11th-power residue symbols in the cyclotomic field \(\mathbb{Q}( \zeta_{11})\), where \(\zeta_{11}\) is a primitive 11th root of unity. It extends an earlier algorithm due to \textit{P. C. Caranay} and \textit{R. Scheidler} [Int. J. Number Theory 6, No. 8, 1831--1853 (2010; Zbl 1242.11079)] for the 7th-power residue symbol. The new algorithm finds applications in the implementation of certain cryptographic schemes.Primes in arithmetic progressions and nonprimitive rootshttps://zbmath.org/1472.110312021-11-25T18:46:10.358925Z"Moree, Pieter"https://zbmath.org/authors/?q=ai:moree.pieter"Sha, Min"https://zbmath.org/authors/?q=ai:sha.minSummary: Let \(p\) be a prime. If an integer \(g\) generates a subgroup of index \(t\) in \((\mathbb{Z}/p\mathbb{Z})^{\ast },\) then we say that \(g\) is a \(t\)-near primitive root modulo \(p\). We point out the easy result that each coprime residue class contains a subset of primes \(p\) of positive natural density which do not have \(g\) as a \(t\)-near primitive root and we prove a more difficult variant.A generalization of regular convolutions and Ramanujan sumshttps://zbmath.org/1472.110322021-11-25T18:46:10.358925Z"Burnett, Joseph Vade"https://zbmath.org/authors/?q=ai:burnett.joseph-vade"Osterman, Otto Vaughn"https://zbmath.org/authors/?q=ai:osterman.otto-vaughnAuthors' abstract: Regular convolutions of arithmetical functions were first defined by \textit{W. Narkiewicz} [Colloq. Math. 10, 81--94 (1963; Zbl 0114.26502)]. Useful identities regarding generalizations of the totient-counting function and Ramanujan sums were later proven for regular convolutions by \textit{P. J. McCarthy} [Port. Math. 27, 1--13 (1968; Zbl 0203.35304)] and \textit{K. N. Rao} [Studies in arithmetical functions. University of Delhi (Ph.D. thesis) (1967)]. We introduce semi-regular convolutions as a generalization of the regular convolutions and show that many of these identities still hold. In particular, special cases of the generalized Ramanujan sums correspond to the corresponding expected generalizations of the totient-counting and Möbius functions. Then we demonstrate that the class of semi-regular convolutions is the broadest generalization to multiplicative-preserving convolutions possible in which even the most basic of these identities still hold. Finally, we introduce a convolution related to Cohen's infinitary convolution [ \textit{G. L. Cohen} and \textit{P. Hagis jun.}, Int. J. Math. Math. Sci. 16, No. 2, 373--383 (1993; Zbl 0774.11006)] that is semi-regular. This convolution has never been studied to the best of the authors' knowledge and possesses a property that distinguishes it from all of the other semi-regular convolutions.Exceptional totient numbershttps://zbmath.org/1472.110332021-11-25T18:46:10.358925Z"Duncan, Breille"https://zbmath.org/authors/?q=ai:duncan.breille"Harrington, Joshua"https://zbmath.org/authors/?q=ai:harrington.joshua"Vincent, Andrew"https://zbmath.org/authors/?q=ai:vincent.andrew-f|vincent.andrew-dSummary: A positive integer \(n\) is called an exceptional totient number if the set \(R_e(n) = \{x \in \mathbb{Z}: 1 \le x < n\), \(\gcd(n, x) = \gcd(n, x - 1) = 1\}\) can be partitioned into two disjoint subsets of equal sum. Take, for example, \(R_e(7) =\{2,3,4,5,6\}\). This can be partitioned into the subsets \(\{2,3,5\}\) and \(\{4,6\}\) whose elements each add to 10. In this article, we provide a complete classification of exceptional totient numbers.Cardinality of a floor function sethttps://zbmath.org/1472.110342021-11-25T18:46:10.358925Z"Heyman, Randell"https://zbmath.org/authors/?q=ai:heyman.randellThe author determines the cardinality of the set $S(X)=\{m:m= [X/n]\text{ for some }n<=X\}$, where $X$ is a positive integer, and $[ ]$ denotes the integer-part (or floor)-function. He also gives lower and upper bounds for the cardinality of the set $S_d(X) =\{m: m=[X/n]\text{ for some } n<=X$ and $d$ divides $[X/n]\}$.Polynomial analogue of the Smarandache functionhttps://zbmath.org/1472.110352021-11-25T18:46:10.358925Z"Li, Xiumei"https://zbmath.org/authors/?q=ai:li.xiumei"Sha, Min"https://zbmath.org/authors/?q=ai:sha.minSummary: In the integer case, the Smarandache function of a positive integer \(n\) is defined to be the smallest positive integer \(k\) such that \(n\) divides the factorial \(k\)!. In this paper, we first define a natural order for polynomials in \(\mathbb{F}_q [t]\) over a finite field \(\mathbb{F}_q\) and then define the Smarandache function of a non-zero polynomial \(f \in \mathbb{F}_q [t]\), denoted by \(S(f)\), to be the smallest polynomial \(g\) such that \(f\) divides the Carlitz factorial of \(g\). In particular, we establish an analogue of a problem of Erdős, which implies that for almost all polynomials \(f, S(f) = t^d\), where \(d\) is the maximal degree of the irreducible factors of \(f\).On the prime factors of the iterates of the Ramanujan \(\tau\)-functionhttps://zbmath.org/1472.110362021-11-25T18:46:10.358925Z"Luca, Florian"https://zbmath.org/authors/?q=ai:luca.florian"Mabaso, Sibusiso"https://zbmath.org/authors/?q=ai:mabaso.sibusiso"Stănică, Pantelimon"https://zbmath.org/authors/?q=ai:stanica.pantelimonThe Ramanujan \( \tau \)-function \( \tau(n) \) is defined as the coefficient of \( q^n \) in the expansion
\begin{align*}
q\prod_{k=1}^{\infty}(1-q^{k})^{24}= \sum_{n\ge 1}\tau(n)q^{n},
\end{align*}
where \( q=\exp(2\pi i z) \), with \( z \) in the complex upper half plane. It is well-known that \( \tau \) is a multiplicative function, that is, \( \tau(1)=1 \) and \( \tau(mn)=\tau(m)\tau(n) \) holds for all relatively prime positive integers \( m,n \). In the paper under review, the authors study the size and all the prime factors of the iterates of \( \tau(n) \), for a positive integer \( n\ge 1 \).Some properties of Zumkeller numbers and \(k\)-layered numbershttps://zbmath.org/1472.110372021-11-25T18:46:10.358925Z"Mahanta, Pankaj Jyoti"https://zbmath.org/authors/?q=ai:mahanta.pankaj-jyoti"Saikia, Manjil P."https://zbmath.org/authors/?q=ai:saikia.manjil-p"Yaqubi, Daniel"https://zbmath.org/authors/?q=ai:yaqubi.danielSummary: Generalizing the concept of a perfect number is a Zumkeller or integer perfect number that was introduced by Zumkeller in 2003. The positive integer \(n\) is a Zumkeller number if its divisors can be partitioned into two sets with the same sum, which will be \(\sigma(n) / 2\). Generalizing even further, we call \(n\) a \(k\)-layered number if its divisors can be partitioned into \(k\) sets with equal sum.
In this paper, we completely characterize Zumkeller numbers with two distinct prime factors and give some bounds for prime factorization in case of Zumkeller numbers with more than two distinct prime factors. We also characterize \(k\)-layered numbers with two distinct prime factors and even \(k\)-layered numbers with more than two distinct odd prime factors. Some other results concerning these numbers and their relationship with practical numbers and Harmonic mean numbers are also discussed.Factorization theorems for relatively prime divisor sums, GCD sums and generalized Ramanujan sumshttps://zbmath.org/1472.110382021-11-25T18:46:10.358925Z"Mousavi, Hamed"https://zbmath.org/authors/?q=ai:mousavi.hamed"Schmidt, Maxie D."https://zbmath.org/authors/?q=ai:schmidt.maxie-dThe authors give new factorization theorems for Lambert series which allow them to express formal generating functions for special sums as invertible matrix transformations involving partition functions.
There are many formulas in the paper which look complicated to the reviewer but the authors of the article believe that the formulas will be useful in the future. Note also that there are two typographical errors on the second page of the paper. First, the last term in equation (1) should be
\[ \sum_{\substack{1<d\leq x \\ 1<(d,x)<d}}f(d). \]
Secondly, the term \(f(15)\) should be added to the right-hand side of the equation for \(\tilde{S_f}(24)\). There might be more errors on the other pages of the paper.Evaluation of convolution sums entailing mixed divisor functions for a class of levelshttps://zbmath.org/1472.110392021-11-25T18:46:10.358925Z"Ntienjem, Ebénézer"https://zbmath.org/authors/?q=ai:ntienjem.ebenezerSummary: Let \(0< n\), \(\alpha,\beta\in\mathbb{N}\) be such that \(\mathrm{gcd} (\alpha,\beta) = 1\). We carry out the evaluation of the convolution sums \[\sum_{\substack{(k,l)\in\mathbb{N}^{2} \\ \alpha\,k+\beta\, l=n}} \sigma (k)\sigma_3(l)\,\,\text{and}\,\, \sum_{\substack{(k,l)\in\mathbb{N}^{2} \\ \alpha\,k+\beta\,l=n}} \sigma_3 (k)\sigma (l)\] for all levels \(\alpha\beta\in\mathbb{N} \), by using in particular modular forms. We next apply convolution sums belonging to this class of levels to determine formulae for the number of representations of a positive integer \(n\) by the quadratic forms in twelve variables \(\sum^{12}_{i=1} x^2_i\) when the level \(\alpha\beta\equiv 0\,\,(\text{mod}\,4)\), and \(\sum_{i=1}^6 (x^2_{2i-1}+x_{2i-1}x_{2i}+x^2_{2i})\) when the level \(\alpha\beta\equiv 0\,\, (\text{mod}\,3)\). Our approach is then illustrated by explicitly evaluating the convolution sum for \(\alpha \beta = 3, 6, 7, 8, 9, 12, 14, 15, 16, 18, 20, 21, 27, 32\). These convolution sums are then applied to determine explicit formulae for the number of representations of a positive integer \(n\) by quadratic forms in twelve variables.On a paper of Dressler and van de Lunehttps://zbmath.org/1472.110402021-11-25T18:46:10.358925Z"Panzone, P. A."https://zbmath.org/authors/?q=ai:panzone.pablo-aSummary: If \(z\in\mathbb{C}\) and \(1\le n\) is a natural number then \[ \sum_{d_1 d_2 =n} (1-z^{p_1})\cdots (1-z^{p_m}) z^{q_1 e_1+\cdots +q_i e_i}=1, \] where \(d_1=p_1^{r_1}\dots p_m^{r_m}, d_2=q_1^{e_1}\dots q_i^{e_i}\) are the prime decompositions of \(d_1, d_2\). This is one of the identities involving arithmetic functions that we prove using ideas from the paper of \textit{R. E. Dressler} and \textit{J. van de Lune} [Proc. Am. Math. Soc. 41, 403--406 (1973; Zbl 0273.10003)].On some new arithmetic properties of the generalized Lucas sequenceshttps://zbmath.org/1472.110412021-11-25T18:46:10.358925Z"Andrica, Dorin"https://zbmath.org/authors/?q=ai:andrica.dorin"Bagdasar, Ovidiu"https://zbmath.org/authors/?q=ai:bagdasar.ovidiu-dumitruSummary: Some arithmetic properties of the generalized Lucas sequences are studied, extending a number of recent results obtained for Fibonacci, Lucas, Pell, and Pell-Lucas sequences. These properties are then applied to investigate certain notions of Fibonacci, Lucas, Pell, and Pell-Lucas pseudoprimality, for which we formulate some conjectures.Biases in prime factorizations and Liouville functions for arithmetic progressionshttps://zbmath.org/1472.110422021-11-25T18:46:10.358925Z"Humphries, Peter"https://zbmath.org/authors/?q=ai:humphries.peter-j"Shekatkar, Snehal M."https://zbmath.org/authors/?q=ai:shekatkar.snehal-m"Wong, Tian An"https://zbmath.org/authors/?q=ai:wong.tian-anLet \(\lambda(n)\) be the Liouville function, that is defined as \(\lambda(n) = (-1)^{\Omega(n)}\) where \(\Omega(n)\) is the total number of prime factors of the positive integer \(n\) (counting its multiplicity). It is well known that the Riemann hypothesis is equivalent to the statement that \(L(x) := \sum_{n \le x} \lambda(n) = O_\varepsilon(x^{1/2+\varepsilon})\) for any \(\varepsilon > 0\), whereas the prime number theorem is equivalent to the estimate \(o(x)\).
This paper introduces some refinements of the Liouville function which detect how primes in given arithmetic progressions appear in prime factorizations, and studies its average order. With this aim, let \(\Omega(n; q, a)\) be the number of prime factors of \(n\) congruent to \(a\) modulo \(q\), and \(\lambda(n; q, a) = (-1)^{\Omega(n; q, a)}\); then, a natural question is to ask for the asymptotic behavior of \(L(x; q, a) := \sum_{n \le x} \lambda(n; q, a)\).
Yet more, the paper defines the \(r\)-fold products
\[
\lambda(n ; q, a_{1}, \dots, a_{r}) = \prod_{i=1}^{r} \lambda(n ; q, a_{i})
\]
where the \(a_i\) are distinct residue classes modulo \(q\), with \(1 \le r \le q\), and define \(\Omega(n;q,a_1,\dots,a_r)\) and \(L(x;q,a_1,\dots,a_r)\) analogously, and studies the order or \(L(x;q,a_1,\dots,a_r)\).
For instance, it is shown that, for \(q \ge 2\) and \(a_1,\dots,a_{\varphi(q)}\) the residue classes modulo \(q\) such that \(\gcd(a_i,q) = 1\), we have
\[
\sum_{n \leq x} \lambda(n ; q, a_{1}, \dots, a_{\varphi(q)}) = o(x)
\]
(and \(O_\varepsilon(x^{1/2+\varepsilon}\) if we assume the Riemann hypothesis). However, if \(b_1,\dots,b_{q-\varphi(q)}\) are the remaining residues classes modulo \(q\), and we take any subset of size \(k\), the asymptotic behavior of \(\sum_{n \leq x} \lambda(n ; q, b_{1}, \dots, b_{k})\) is different of what happens with \(a_{1}, \dots, a_{\varphi(q)}\).
Moreover, supported by numerical tests, the paper analyzes analogues of Pólya's conjecture (i.e., that \(L(x) \le 0\) for \(x \ge 2\), that where disproved by Haselgrove in 1958), and proves results related to the sign changes of the associated summatory functions.Architecture of inflorescences and continued fractionshttps://zbmath.org/1472.110432021-11-25T18:46:10.358925Z"Casares Antón, Carmen"https://zbmath.org/authors/?q=ai:casares-anton.carmenTranslation from the Spanish: The seeds of a sunflower trace out families of spiral branches in a number equal to different terms of the Fibonacci sequence. But, is this arrangement the only solution to the evolutionary problem of the distribution of seeds in botanical species? The continued fractions associated with the real numbers and their congruences offer an answer to this question. In this article, we show arguments that dismiss some architectures of inflorescences and highlight others, different from Fibonacci, as options that nature has also developed.The Möbius transformation of continued fractions with bounded upper and lower partial quotientshttps://zbmath.org/1472.110442021-11-25T18:46:10.358925Z"Liu, Wencai"https://zbmath.org/authors/?q=ai:liu.wencaiSummary: Let \(h: x \mapsto\frac{ax+b}{cx+d}\) be the nondegenerate Möbius transformation with integer entries. We get a bound of the continued fraction of \(h(x)\) by upper and lower bounds of the continued fraction of \(x\).On the minimal Hamming weight of a multi-base representationhttps://zbmath.org/1472.110452021-11-25T18:46:10.358925Z"Krenn, Daniel"https://zbmath.org/authors/?q=ai:krenn.daniel"Suppakitpaisarn, Vorapong"https://zbmath.org/authors/?q=ai:suppakitpaisarn.vorapong"Wagner, Stephan"https://zbmath.org/authors/?q=ai:wagner.stephan-gSummary: Given a finite set of bases \(b_1, b_2, \ldots, b_r\) (integers greater than 1), a multi-base representation of an integer \(n\) is a sum with summands \(db_1^{\alpha_1} b_2^{\alpha_2} \cdots b_r^{\alpha_r}\), where the \(\alpha_j\) are nonnegative integers and the digits \(d\) are taken from a fixed finite set. We consider multi-base representations with at least two bases that are multiplicatively independent. Our main result states that the order of magnitude of the minimal Hamming weight of an integer \(n\), i.e., the minimal number of nonzero summands in a representation of \(n\), is \(\log n /(\log \log n)\). This is independent of the number of bases, the bases themselves, and the digit set.
For the proof, the existing upper bound for prime bases is generalized to multiplicatively independent bases; for the required analysis of the natural greedy algorithm, an auxiliary result in Diophantine approximation is derived. The lower bound follows by a counting argument and alternatively by using communication complexity; thereby improving the existing bounds and closing the gap in the order of magnitude. This implies also that the greedy algorithm terminates after \(\mathcal{O}(\log n / \log \log n)\) steps, and that this bound is sharp.Permutative numbershttps://zbmath.org/1472.110462021-11-25T18:46:10.358925Z"Lucht, Lutz G."https://zbmath.org/authors/?q=ai:lucht.lutz-gerhard"Motzer, Renate"https://zbmath.org/authors/?q=ai:motzer.renateZusammenfassung: Zur Untersuchung des 2017/18 von \textit{H. Hischer} [GDM Mitteil. 105, 29--30 (2018)] kommunizierten Vorkommens von 10-stelligen Dezimalzahlen, die sämtliche Ziffern \(1,2,\ldots,9\) enthalten und zu denen es einen ganzzahligen Teiler \(>1\) derart gibt, daß die Division nur ihre Ziffernreihenfolge verändert, wird hier der Begriff der permutativen Zahl eingeführt. Dieser Beitrag liefert einige allgemeine Ergebnisse über Existenz und Konstruktion permutativer Zahlen und weist auf offene Probleme hin.On the distribution of the sum of digits of sums \(a+b\)https://zbmath.org/1472.110472021-11-25T18:46:10.358925Z"Mauduit, Christian"https://zbmath.org/authors/?q=ai:mauduit.christian"Rivat, Joël"https://zbmath.org/authors/?q=ai:rivat.joel"Sárközy, András"https://zbmath.org/authors/?q=ai:sarkozy.andrasFor \(N>2\) let \(\mathcal{A}, \mathcal{B}\) be subsets of \(\{1,2, \ldots,N\}\), and denote by \(s(n)\) the binary sum of digits of the integer \(n\). The authors investigate the distribution of \(s(a+b)\) for \((a,b)\in \mathcal{A}\times \mathcal{B}\). They prove that, as \(N\rightarrow \infty\),
\begin{align*}
\frac{1}{|\mathcal{A}| |\mathcal{B}|} &\left |\{(a,b)\in \mathcal{A}\times \mathcal{B},\;\; s(a+b)<z\}\right|\\
&=\Phi(y_{2N})+O\left(\frac{N}{\sqrt{|\mathcal{A}| |\mathcal{B}|}} \frac{(\log \log N) (\log \log \log N)^{1/2}}{(\log N)^{1/4}}\right),
\end{align*}
uniformly in \(z\) with \(0<z<\log(2N)/\log 2\), where \(\Phi(u)=(2\pi)^{-1/2}\int_{-\infty}^u e^{-t^2/2} dt\), and \(y_{2N}=y_{2N}(z)\) is defined by \[z=\frac{\log 2N}{\log 4}+y_{2N} \cdot \frac{1}{2}\left(\frac{\log 2N}{\log 2}\right)^{1/2}.\] As the authors note, this result implies that, as long as \[\sqrt{|\mathcal{A}| |\mathcal{B}|}\cdot \frac{(\log N)^{1/4}}{N \log \log N} \rightarrow \infty,\quad N\rightarrow \infty,\] the distribution of \(s(a+b)\) for \((a,b) \in \mathcal{A}\times \mathcal{B}\) resembles the distribution of \(s(n)\) for \(n\leq 2N\).Zeta function and negative beta-shiftshttps://zbmath.org/1472.110482021-11-25T18:46:10.358925Z"Ndong, Florent Nguema"https://zbmath.org/authors/?q=ai:nguema-ndong.florentSummary: Given a real number \( \beta > 1\), we study the associated \( (-\beta )\)-shift introduced by \textit{S. Ito} and \textit{T. Sadahiro} [Integers 9, No. 3, 239--259, A22 (2009; Zbl 1191.11005)]. We compare some aspects of the \((-\beta )\)-shift to the \(\beta \)-shift. When the expansion in base \( -\beta \) of \( -\frac{\beta }{\beta +1} \) is periodic with odd period or when \( \beta \) is less than the golden ratio, the \( (-\,\beta )\)-shift cannot be coded because its language is not transitive. This intransitivity of words explains the existence of gaps in the interval \([-\frac{\beta }{\beta +1}, \frac{1}{\beta +1})\). We observe that an intransitive word appears in the \((-\,\beta )\)-expansion of a real number taken in the gap. Furthermore, we determine the Zeta function \(\zeta _{-\beta }\) of the \((-\beta )\)-transformation and the associated lap-counting function \(L_{T_{-\beta }}\). These two functions are related by \(\zeta _{-\beta }=(1-z^2)L_{T_{-\beta }}\). We observe some similarities with the zeta function of the \(\beta \)-transformation. The function \(\zeta _{-\beta }\) has a simple pole at \( \frac{1}{\beta }\) and no other singularities \(z\) such that \(| z | =\frac{1}{\beta }\). We also note an influence of gaps (\(\beta \) less than the golden ratio) on the zeta function.Approaching Cusick's conjecture on the sum-of-digits functionhttps://zbmath.org/1472.110492021-11-25T18:46:10.358925Z"Spiegelhofer, Lukas"https://zbmath.org/authors/?q=ai:spiegelhofer.lukasLet \(s(n)\) denote the sum of the digits in \(n\) when expressed in binary. Cusick's conjecture claims that \(c_t \geq \frac 12\) for all \(t\geq 0\), where \(c_t\) stands for the natural density of the set of all \(n\geq 0\) such that \(s(n+t)\geq s(n)\). The conjecture arose in connection with the work of \textit{Z. Tu} and \textit{Y. Deng} [Des. Codes Cryptography 60, No. 1, 1--14 (2011; Zbl 1226.94013)] on finding Boolean functions satisfying certain cryptographical conditions.
Letting \(t'=3\cdot 2^{\lambda}-t\), with \(2^{\lambda}\leq t\leq 2^{\lambda+1}\), the author shows that for every \(\varepsilon \geq 0\), there exists \(C \geq 0\) such that \(c_t+c_{t'}\geq 1-\varepsilon\) whenever the digits in \(t\) have at least \(C\) blocks of consecutive 1s. With such knowledge, we can construct many examples of \(t\) for which \(c_t\geq \frac 12-\varepsilon\), and that seems encouraging considering that Cusick's conjecture has remained open despite having been previously shown to be asymptotically true for almost all \(t\).Characterization and enumeration of palindromic numbers whose squares are also palindromichttps://zbmath.org/1472.110502021-11-25T18:46:10.358925Z"Tripathi, Amitabha"https://zbmath.org/authors/?q=ai:tripathi.amitabhaSummary: Palindromic numbers are positive integers that remain unchanged when their decimal digits are reversed. We characterize palindromic numbers whose squares are also palindromic. We use this to determine the number of \(n\)-digit palindromic numbers whose squares are palindromic, and the number of palindromic numbers whose squares are palindromic and which are not greater than a fixed positive integer.Fibonacci numbers which are concatenations of two repdigitshttps://zbmath.org/1472.110672021-11-25T18:46:10.358925Z"Alahmadi, Adel"https://zbmath.org/authors/?q=ai:alahmadi.adel-n"Altassan, Alaa"https://zbmath.org/authors/?q=ai:altassan.alaa"Luca, Florian"https://zbmath.org/authors/?q=ai:luca.florian"Shoaib, Hatoon"https://zbmath.org/authors/?q=ai:shoaib.hatoonSummary: We show that the only Fibonacci numbers that are concatenations of two repdigits are 13, 21, 34, 55, 89, 144, 233, 377.A generalization of primitive sets and a conjecture of Erdőshttps://zbmath.org/1472.110772021-11-25T18:46:10.358925Z"Chan, Tsz Ho"https://zbmath.org/authors/?q=ai:chan.tsz-ho"Lichtman, Jared Duker"https://zbmath.org/authors/?q=ai:lichtman.jared-duker"Pomerance, Carl"https://zbmath.org/authors/?q=ai:pomerance.carlA set of integers greater than 1 is primitive, if no element of the set divides another element of the set. Erdős proved in 1935 that there is a number \(K\) such that for every primitive set \(A\), \(\sum_{n\in A} 1/n \leq K\). In 1988 Erdős conjectured that the least upper bound \(K\) is realized by the set of primes. There is a generalization of primitivity: a set \(A\) of integers greater than 1 with \(|A|\geq k+1\) is called \(k\)-primitive, if no element of \(A\) divides the product of \(k\) distinct other elements of \(A\). The paper under review shows that the sum of reciprocals of the elements of any 2-primitive set up to any bound \(n\) is at most the sum of reciprocals of primes up to the bound \(n\), a nice analogue of the conjecture of Erdős.An extension of a formula of Jovovichttps://zbmath.org/1472.110782021-11-25T18:46:10.358925Z"Chern, Shane"https://zbmath.org/authors/?q=ai:chern.shane\textit{P. Erdős} et al. [Bull. Res. Council Israel 10F, 41--43 (1961; Zbl 0063.00009)] proves that for any set \(S\) of \(2n-1\) integers, there exists a subset of \(n\) elements whose sum is divisible by \(n\). When \(S=\{1,2,\ldots, 2n-1\}\), the number of such subsets is given by \[ \frac{(-1)^n}{2n}\sum_{d\mid n}(-1)^d \phi\left(\frac nd\right)\binom{2d}{d}, \] an expression conjectured by Jovovic and proved by Alekseyev [\url{https://oeis.org/A145855}].
In this article, the author generalizes this result via a new combinatorial approach and gives a closed form of the number of subsets of \(\{1,2,\ldots, un-1\}\) of \(vn\) elements whose sum is congruent to \(c\) modulo \(n\) (Theorem 1.1).Linear combinations of two polygonal numbers that take infinitely often a square valuehttps://zbmath.org/1472.110992021-11-25T18:46:10.358925Z"Li, Yangcheng"https://zbmath.org/authors/?q=ai:li.yangchengThe main objective of this paper is to establish certain sufficient conditions on the multipliers of two distinct polygonal numbers, which when multiplied and then addup to a square. Mathematically, if \(P_p (x)\) and \(P_q (y)\) denote the \(x\)-th \(p\)-gonal and \(y\)-th \(q\)-gonal number respectively, then there are certain sufficient conditions on the positive integers \(m\) and \(n\) such that the Diophantine equation \[mP_p(x)+nP_q(y)=z^2,\quad p\geq 3,~q\geq 3\] have infinitely many positive integer solutions. To prove the main results of this work, the theory of congruence and the properties of Pell equation have been used.This work not only extends the earlier study of the case \(Ax^2+By^2=Cz^2\), but also some more studies like squares whiuch are linear combination of two \(g-\)gonal numbers, squares which differ from fixed multiples of triangular numbers by one, etc. by several authors from time to time.Analogues of the Robin-Lagarias criteria for the Riemann hypothesishttps://zbmath.org/1472.112362021-11-25T18:46:10.358925Z"Washington, Lawrence C."https://zbmath.org/authors/?q=ai:washington.lawrence-c"Yang, Ambrose"https://zbmath.org/authors/?q=ai:yang.ambroseLet \(\sigma(n)=\sum_{d\mid n}d\) be the sum of divisors function. In [J. Math. Pures Appl., Sér. 63, 187--213 (1984; Zbl 0516.10036)] \textit{G. Robin} has shown that the Riemann hypothesis (RH, in short) is equivalent to \(\sigma(n)<e^{\gamma}n\log\log n\) for any \(n>5040\), where \(\gamma\) is the Euler-Mascheroni constant. In [Am. Math. Mon. 109, 534--543 (2002; Zbl 1098.11005)] \textit{J. C. Lagarias} has reformulated Robin's criterion for RH. Namely, he proved that RH is equivalent to \(\sigma(n)\leq H_n+\exp(H_n)\log(H_n)\) for any \(n\geq 1\), where \(H_n\) is the \(n\)-th harmonic number.
In this paper, the authors show that RH is equivalent to \[\sigma(n)<\frac{e^{\gamma}}{2}n\log\log n\] for any odd integers \(n\) with \(n\geq 3^4\cdot 5^3\cdot 7^2\cdot \prod_{11\leq p\leq 67,\text{ primes}}p\). This can be regarded as an analogue of Robin's criterion for RH. The authors also give that RH is equivalent to an inequality that \(\sigma(n)\) is bounded above by a quantity in terms of an analogue of the harmonic number for any odd \(n\). The authors also prove that the inequality \(\sigma(n)<e^{\gamma}n\log\log n/2\) holds for any sufficiently large odd integers \(n\) belonging to certain sets of integers. For example, the inequality is true when \(n\) are sufficiently large odd square-free integers.
For the proof of the analogue of Robin's criterion the authors introduce the concept of odd colossally abundant numbers, which is an analogue of the concept of colossally abundant numbers by \textit{L. Alaoglu} and \textit{P. Erdős} [Trans. Am. Math. Soc. 56, 448--469 (1944; Zbl 0061.07903)].The completed finite period map and Galois theory of supercongruenceshttps://zbmath.org/1472.112382021-11-25T18:46:10.358925Z"Rosen, Julian"https://zbmath.org/authors/?q=ai:rosen.julianSummary: A period is a complex number arising as the integral of a rational function with algebraic number coefficients over a region cut out by finitely many inequalities between polynomials with rational coefficients. Although periods are typically transcendental numbers, there is a conjectural Galois theory of periods coming from the theory of motives. This paper formalizes an analogy between a class of periods called multiple zeta values and congruences for rational numbers modulo prime powers (called supercongruences). We construct an analog of the motivic period map in the setting of supercongruences and use it to define a Galois theory of supercongruences. We describe an algorithm using our period map to find and prove supercongruences, and we provide software implementing the algorithm.Correction to: ``Gaps between divisible terms in \(a^2(a^2 + 1)\)''https://zbmath.org/1472.112482021-11-25T18:46:10.358925Z"Chan, Tsz Ho"https://zbmath.org/authors/?q=ai:chan.tsz-hoCorrection to the author's paper [ibid. 101, No. 3, 396--400 (2020; Zbl 1468.11196)].Sums of logarithmic averages of gcd-sum functionshttps://zbmath.org/1472.112562021-11-25T18:46:10.358925Z"Kiuchi, Isao"https://zbmath.org/authors/?q=ai:kiuchi.isao"Eddin, Sumaia Saad"https://zbmath.org/authors/?q=ai:saad-eddin.sumaiaSummary: Let \(\gcd (k,j)\) be the greatest common divisor of the integers \(k\) and \(j\). For any arithmetic function \(f\), we establish several asymptotic formulas for weighted averages of gcd-sum functions with weight concerning logarithms, that is \[\sum\limits_{k \le x} \frac{1}{k} \sum\limits_{j = 1}^k f(\gcd (k,j)) \log j.\] More precisely, we give asymptotic formulas for various multiplicative functions such as \(f = \text{id}, \phi, \text{id}_{1+a}\) and \(\phi_{1+a}\) with \(-1 < a < 0\). We also consider several formulas of Dirichlet series associated with Anderson-Apostol sums.Sums of weighted averages of gcd-sum functions. IIhttps://zbmath.org/1472.112572021-11-25T18:46:10.358925Z"Kiuchi, Isao"https://zbmath.org/authors/?q=ai:kiuchi.isao"Eddin, Sumaia Saad"https://zbmath.org/authors/?q=ai:saad-eddin.sumaiaSummary: We establish two identities involving the Gamma function and Bernoulli polynomials, namely \[\sum_{k \leq x} \frac{1}{k^{s}} \sum_{j=1}^{k^{s}} \log \Gamma \bigg(\frac{j}{k^{s}}\bigg) \sum_{d|k,d^s|j} f \ast \mu (d) \quad\text{and}\quad \sum_{k \leq x} \frac{1}{k^{s}} \sum_{j = 0}^{k^s-1} B_m \sum_{d|k,d^{s}|j} f \ast \mu (d)\] with any fixed integer \(s > 1\) and any arithmetical function \(f\). We give asymptotic formulas for them with various multiplicative functions \(f\). We also consider several formulas of Dirichlet series associated with the above identities. This paper is a continuation of an earlier work of the authors.
For part I see Result. Math. 75, No. 2, Paper No. 53, 22 p. (2020; Zbl 1472.11256).Explicit bounds for small prime nonresidueshttps://zbmath.org/1472.112592021-11-25T18:46:10.358925Z"Ma, Shilin"https://zbmath.org/authors/?q=ai:ma.shilin"McGown, Kevin"https://zbmath.org/authors/?q=ai:mcgown.kevin-j"Rhodes, Devon"https://zbmath.org/authors/?q=ai:rhodes.devon"Wanner, Mathias"https://zbmath.org/authors/?q=ai:wanner.mathiasSummary: Let \(\chi\) be a Dirichlet character modulo a prime \(p\). We give explicit upper bounds on \(q_1 < q_2 < \cdots < q_n\), the \(n\) smallest prime nonresidues of \(\chi\). More precisely, given \(n_0\) and \(p_0\) there exists an absolute constant \(C = C(n_0, p_0) > 0\) such that \(q_n \leq C p^{\frac{1}{4}}(\log p)^{\frac{n + 1}{2}}\) whenever \(n \leq n_0\) and \(p \geq p_0\).Asymptotic behavior of functions \(\Omega(k;n)\) and \(\omega(k;n)\) related to the number of prime divisorshttps://zbmath.org/1472.112622021-11-25T18:46:10.358925Z"Shubin, Andrei V."https://zbmath.org/authors/?q=ai:shubin.andrei-vSummary: This article is related to the average estimates of numerical functions \(\Omega(k;n)\) and \(\omega(k;n)\) connected with the number of prime divisors of \(n\) with limited multiplicity.Counting fixed points and rooted closed walks of the singular map \(x \mapsto x^{x^n}\) modulo powers of a primehttps://zbmath.org/1472.112922021-11-25T18:46:10.358925Z"Holden, Joshua"https://zbmath.org/authors/?q=ai:holden.joshua-brandon"Richardson, Pamela A."https://zbmath.org/authors/?q=ai:richardson.pamela-a"Robinson, Margaret M."https://zbmath.org/authors/?q=ai:robinson.margaret-mSummary: The ``self-power'' map \(x \longmapsto x^x\) modulo \(m\) and its generalized form \(x \mapsto x^{x^n}\) modulo \(m\) are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use \(p\)-adic methods, primarily \(p\)-adic interpolation, Hensel's lemma, and lifting singular points modulo \(p\), to count fixed points and rooted closed walks of equations related to these maps when \(m\) is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacobian matrix.Hausdorff dimension of frequency sets of univoque sequenceshttps://zbmath.org/1472.370162021-11-25T18:46:10.358925Z"Li, Yao-Qiang"https://zbmath.org/authors/?q=ai:li.yaoqiangSummary: We study the set \(\Gamma\) consisting of univoque sequences, the set \(\Lambda\) consisting of sequences in which the lengths of consecutive zeros and consecutive ones are bounded, and their frequency subsets \(\Gamma_a\), \(\underline{\Gamma}_a\), \(\overline{\Gamma}_a\) and \(\Lambda_a\), \(\underline{\Lambda}_a\), \(\overline{\Lambda}_a\) consisting of sequences respectively in \(\Gamma\) and \(\Lambda\) with frequency, lower frequency and upper frequency of zeros equal to some \(a\in[0,1]\). The Hausdorff dimension of all these sets are obtained by studying the dynamical system \((\Lambda^{(m)},\sigma)\) where \(\sigma\) is the shift map and \(\Lambda^{(m)}=\left\{w\in\{0,1\}^{\mathbb{N}}:w\text{ does not contain }0^m\text{ or }1^m\right\}\) for integer \(m\geq 3\), studying the Bernoulli-type measures on \(\Lambda^{(m)}\) and finding out the unique equivalent \(\sigma\)-invariant ergodic probability measure.