Recent zbMATH articles in MSC 11Dhttps://zbmath.org/atom/cc/11D2022-07-25T18:03:43.254055ZWerkzeugOn the Frobenius conjecturehttps://zbmath.org/1487.110312022-07-25T18:03:43.254055Z"Lapointe, Mélodie"https://zbmath.org/authors/?q=ai:lapointe.melodie"Reutenauer, Christophe"https://zbmath.org/authors/?q=ai:reutenauer.christopheMarkoff's equation is a Diophantine equation of the form \(x^2 + y^2 + z^2 = 3xyz,\) and the positive integer solutions \((x,y,z)\) are called Markoff triples. A number is called a Markoff number if it is the maximum of a Markoff triple. The Frobenius conjecture, open for many decades, states that every Markoff number is the maximum of a unique Markoff triple.
Recently several authors have found intriguing connections between this conjecture and Christoffel words; in particular, see the book of the second author [From Christoffel words to Markoff numbers. Oxford: Oxford University Press (2019; Zbl 1443.11002)]. In this interesting paper, the authors generalize the Frobenius conjecture to the uniqueness of the values of a certain map on Sturmian words, and they obtain some partial uniqueness results along these lines.
Reviewer: Jeffrey Shallit (Waterloo)What positive integers \(n\) can be presented in the form \(n=(x+y+z)(1/x+1/y+1/z)\)?https://zbmath.org/1487.110322022-07-25T18:03:43.254055Z"Nguyen Xuan Tho"https://zbmath.org/authors/?q=ai:nguyen-xuan-tho.Summary: This paper shows that the equation in the title does not have positive integer solutions when \(n\) is divisible by 4. This gives a partial answer to a question by Melvyn Knight, see \textit{A. Bremner} et al. [Math. Comput. 61, No. 203, 117--130 (1993; Zbl 0808.11022)]. The proof is a mixture of elementary \(p\)-adic analysis and elliptic curve theory.Diophantine equations with balancing-like sequences associated to Brocard-Ramanujan-type problemhttps://zbmath.org/1487.110332022-07-25T18:03:43.254055Z"Sahukar, Manasi Kumari"https://zbmath.org/authors/?q=ai:sahukar.manasi-kumari"Panda, Gopal Krishna"https://zbmath.org/authors/?q=ai:panda.gopal-krishnaSummary: In this paper, we deal with the Brocard-Ramanujan-type equations \(A_{n_1} A_{n_2} \cdots A_{n_k} \pm 1 = A_m\) or \(G_m\) or \(G_m^2\) where \(\{A_n\}_{n \geq 0}\) and \(\{G_m\}_{m \geq 0}\) are either balancing-like sequences or associated balancing-like sequences.The Diophantine equation \(x^2+3^a\cdot 5^b\cdot 11^c=4y^n\)https://zbmath.org/1487.110342022-07-25T18:03:43.254055Z"Nguyen Xuan Tho"https://zbmath.org/authors/?q=ai:nguyen-xuan-tho.From the text: We investigate the Diophantine equation (*) \(x^2+3^a\cdot 5^b\cdot 11^c=4y^n\) with \(n\ge 3,x,y,a,b,c,d\in \mathbb{N}, x,y>0\), and \(\gcd(x,y)=1\). All integer solutions to the equation (*) are given in Tables 1, 4, 5, 7, and 8.
The main tool is the so-called primitive divisor theorem of Lucas numbers by Bilu, Hanrot, and Voutier [\textit{Yu. Bilu} et al., J. Reine Angew. Math. 539, 75--122 (2001; Zbl 0995.11010)].Smooth values of quadratic polynomialshttps://zbmath.org/1487.110852022-07-25T18:03:43.254055Z"Conrey, J. B."https://zbmath.org/authors/?q=ai:conrey.john-brian"Holmstrom, M. A."https://zbmath.org/authors/?q=ai:holmstrom.m-aLet \(\mathcal{Q}_a(z)=\{q \,;\, p|(q^2+a) \Rightarrow p\leq z\}\), i.e., \(\mathcal{Q}_a(z)\) is the set of \(z\)-smooth numbers of the form \(q^2+a\). It is known that \(\mathcal{Q}_a(z)\) is a finite set for \(a\neq 0\). When \(a=\pm 1, \pm 2, \pm 4\), then \(\mathcal{Q}_a(z)\) can in principle be found by the Pell equation method. In this paper, the authors describe an algorithm which for each \(a\) and \(z\) leads to a subset \(Q_a(z) \subset \mathcal{Q}_a(z)\) that can be calculated quickly and for which they conjecture that \(|Q_a(z)| = |\mathcal{Q}_a(z)|(1+o_a(1))\). Analyzing these sets has led the authors to several conjectures. One of them is that the set of logarithms of the elements of \(Q_a(z)\) become normally distributed for any fixed \(a\) as \(z\to\infty\).
Reviewer: Andrej Dujella (Zagreb)Distribution of values of quadratic forms at integral pointshttps://zbmath.org/1487.110912022-07-25T18:03:43.254055Z"Buterus, P."https://zbmath.org/authors/?q=ai:buterus.paul"Götze, F."https://zbmath.org/authors/?q=ai:gotze.friedrich-w"Hille, T."https://zbmath.org/authors/?q=ai:hille.thomas"Margulis, G."https://zbmath.org/authors/?q=ai:margulis.gregory-aThe problem of understanding the distribution of values of quadratic forms at integer points has been studied from many different perspectives, from the circle method, to Fourier analysis, to, famously, homogeneous dynamics and ergodic theory via the work of Raghunathan, Margulis, Dani-Margulis, and Eskin-Margulis-Mozes among others. This comprehensive paper gives state of the art results in many different situations combining an impressive array of techniques drawn from ergodic theory, dynamics, analysis, and classical lattice point counting.
Authors' abstract: The number of lattice points in \(d\)-dimensional hyperbolic or elliptic shells \(\{m : a<Q[m]<b\}\), which are restricted to rescaled and growing domains \(r\,\Omega \), is approximated by the volume. An effective error bound of order \(o(r^{d-2})\) for this approximation is proved based on Diophantine approximation properties of the quadratic form \(Q\). These results allow to show effective variants of previous non-effective results in the quantitative Oppenheim problem and extend known effective results in dimension \(d \ge 9\) to dimension \(d \ge 5\). They apply to wide shells when \(b-a\) is growing with \(r\) and to positive definite forms \(Q\). For indefinite forms they provide explicit bounds (depending on the signature or Diophantine properties of \(Q)\) for the size of non-zero integral points \(m\) in dimension \(d\ge 5\) solving the Diophantine inequality \(|Q[m] |< \varepsilon\) and provide error bounds comparable with those for positive forms up to powers of \(\log r\).
Reviewer: Jayadev Athreya (Seattle)Restricted partitions of positive integershttps://zbmath.org/1487.110922022-07-25T18:03:43.254055Z"Chen, Jia-Yu"https://zbmath.org/authors/?q=ai:chen.jiayuSummary: For \(a,b,c\in\mathbb{Z}^+=\{1,2,3,\dots\}\) with \(b<c\) and gcd\((b,c)=1\), we show that any integer \(n\ge N(a,b,c)\) can be written as \(x+y+z(x,y,z\in\mathbb{Z}^+)\) with \(ax+(a+b)y+(a+c)z\) a square, where \(N(a,b,c)\) is an explicit expression in terms of \(a,b,c\). For integers \(1<a<c\), we prove that any integer \(n\ge c+2a=1+ \lceil\frac {2a^2-a-1}{c-a}\rceil\) can be written as \(x+y+z(x,y,z\in \mathbb{Z}^+)\) with \(x+2y+cz\) a power of \(a\). Our general results imply two conjectures of Sun Z.-W.The density of fibres with a rational point for a fibration over hypersurfaces of low degreehttps://zbmath.org/1487.140582022-07-25T18:03:43.254055Z"Sofos, Efthymios"https://zbmath.org/authors/?q=ai:sofos.efthymios"Visse-Martindale, Erik"https://zbmath.org/authors/?q=ai:visse-martindale.erikSummary: We prove asymptotics for the proportion of fibres with a rational point in a conic bundle fibration. The base of the fibration is a general hypersurface of low degree.