Powers in arithmetic progressions
type: Item
instance of: scholarly article
title: Powers in arithmetic progressions
zbMATH Open document ID: 1481.11014
author: Lajos Hajdu
author: Szabolcs Tengely
published in: The Ramanujan Journal
publication date: 2021-08-19T00:00:00Z
review text: The authors analyse the function \(P_{a,b;N}(l)\), for integers \(a,b,l\) with \(a>0,l\geq 2\), which enumerates the \(l^{th}\)-powers among the first \(N\) terms \(b,b+a,\cdots,b+(N-1)a\) of the arithmetic progression \(ax+b, x\geq 0\). If \(P_N(l)\) denotes the maximum of these values taken over all arithmetic progressions \(ax+b\), then a conjecture of W.~Rudin predicts that \(P_N(2)=O(\sqrt{N})\). Let \[ S_{a,b}(l)=\underset{N\rightarrow\infty}{\lim}\frac{\mid \{x:ax+b \text{ is an }l^{th} \text{power },0\leq x<N\}\mid }{\sqrt[l]{N}}. \] Let \(S_a(l)=\underset{b\in \mathbb{Z}}{\max}\ S_{a,b}(l)\) and \(S(l)=\underset{a\in \mathbb{N}}{\max}\ S_a(l)\). The \(l\)-transformation of a progression \(ax+b\) is defined as the arithmetic progression \((az^l)x+(b+ta)z^l\) for some positive integer \(z\) and \(t\) an integer. They prove the following main theorem in the article. \(S(l)\) exists for any \(l\geq 2\) and we have \[ S(l)=\begin{cases} \sqrt{\frac 83} & \text{ if }l=2,\\ \underset{\overset{p\text{ prime, } p-1\mid l}{\frac{\log\ p}{\log\ p-\log(p-1)}>l}}{\prod} (p-1)p^{\frac 1l-1} & \text{ otherwise.} \end{cases} \] Further, for the arithmetic progression \(ax+b\) we have \(S_{a,b}(l)=S(l)\) if and only if it is an \(l\)-transformation of \(a^*x+b^*\) where \(a^*=2\) if \(l=2\), \(a^*=5\) or \(80\) if \(l=4\), \(a^*=\underset{\overset{p\text{ prime, } p-1\mid l}{\frac{\log\ p}{\log\ p-\log(p-1)}>l}}{\prod} p\) otherwise and \(b^*=0\) if \(a^*=1\), \(b^*=1\) otherwise. Based on the main theorem, they suggest two problems in the article which are extensions of Rudin's conjecture to the case of general exponents \(l\). It is mentioned that the answer to these two questions proposed is affirmative. To support this, they provide numerical results for \(l=3\) up to \(N<20\) and for \(l=4\) up to \(N<6\) using elliptic and higher genus curves. The affirmative answers to the two proposed questions would give that \(\underset{a>0,b\in \mathbb{Z}}{\max}\ P_{a,b;N}(l)=O(\…
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 11N64
Mathematics Subject Classification ID: 11G30
Mathematics Subject Classification ID: 11D25
zbMATH DE Number: 7383300
zbMATH Keywords: arithmetic progression
zbMATH Keywords: perfect powers
zbMATH Keywords: Rudin's conjecture
reviewed by: C. P. Anil Kumar
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: s11139-020-00331-5
OpenAlex ID: W3127260892
cites work: A note on squares in arithmetic progressions. II
cites work: Squares in arithmetic progressions
cites work: The Magma algebra system. I: The user language
cites work: POWERS FROM PRODUCTS OF CONSECUTIVE TERMS IN ARITHMETIC PROGRESSION
cites work: Q5779949
cites work: Über die diophantische Gleichung \(x^l+y^l=c z^l\)
cites work: Q5726744
cites work: Roots of unity and nullity modulo $n$
cites work: On a conjecture of Rudin on squares in arithmetic progressions
cites work: Squares in Arithmetic Progressions and Infinitely Many Primes
cites work: Q4001741
cites work: Q3265794
cites work: Q4210105
cites work: On the height constant for curves of genus two
cites work: Implementing 2-descent for Jacobians of hyperelliptic curves
cites work: On the height constant for curves of genus two, II
cites work: Q4079680
DOI: 10.1007/S11139-020-00331-5
description: scientific article; zbMATH DE number 7383300