Prime divisors of \(a^n-b^n\)
type: Item
instance of: scholarly article
title: Prime divisors of \(a^n-b^n\)
author: Pradipto Banerjee
published in: Integers
publication date: 2024-05-30T00:00:00Z
review text: Consider the sequence \(u_n=a^n-b^n\) with given integers \(a>b\geq 1\), and let \(P(u_n)\) denote the largest prime factor of this \(n\)-th term.\N\NBy the uniqueness of prime factorization, we express \(u_n=UV^{k+1}\) such that each prime factor of \(U\) has at most \(k\) multiplicity. The author states the following hypothesis: Assume that there exists an absolute constant \(\lambda\) such that \(V<(aU)^{\lambda}\) whenever \(k\geq \lambda\). Then it follows that \(P(u_n)\gg (n/\tau(n))^2\) with an absolute implied constant. Here, \(\tau(n)\) is the number of distinct positive divisors of \(n\).\N\NIn multiple ways, this claim is a clear improvement of earlier results in studying the lower-bounds of \(P(u_n)\). And the stated hypothesis above is itself a weaker condition related to an \(abc\)-conjecture form that has played a role in existing literature.
reviewed by: Amin Witno
Mathematics Subject Classification ID: 11A05
zbMATH DE Number: 7856124
zbMATH Keywords: largest prime factor
zbMATH Keywords: abc-conjecture
MaRDI profile type: Publication
cites work: Resultants of Cyclotomic Polynomials
cites work: Q5341879
cites work: Q4826661
cites work: Q3518705
cites work: The large sieve
cites work: Q4664225
cites work: VARIATIONS ON A THEME OF ROMANOFF
cites work: Q4431809
cites work: Prime divisors of sparse values of cyclotomic polynomials and Wieferich primes
cites work: The greatest prime factor of $a^n - b^n$
cites work: On Divisors of Fermat, Fibonacci, Lucas, and Lehmer Numbers
cites work: On divisors of Lucas and Lehmer numbers
description: scientific article; zbMATH DE number 7856124