A note on the paper by Bugeaud and Laurent “Minoration effective de la distance p-adique entre puissances de nombres algébriques”

Abstract

We shall make a slight improvement to a result of p-adic logarithms, which gives a nontrivial upper bound for the exponent of p dividing the Fermat quotient x p−1 − 1.where p is a prime, α 1 , α 2 are integers not divisible by p and b 1 , b 2 are integers with gcd(b 1 , b 2 , p) = 1. This result has been refined by several papers such as Schinzel [10], Yu [13][14][15][16][17], Bugeaud [2], and Bugeaud and Laurent [1]. Our purpose is to improve a result in the last paper by ✩ This paper is a revised version of my master thesis (Yamada, 2006 [12]).

On divisors of Lucas and Lehmer numbers

“…Yamada [46], by making use of a refinement of an estimate of Bugeaud and Laurent [7] for linear forms in two p-adic logarithms, proved that there is a positive number C 2 , which is effectively computable in terms of ω(a), such that ord p (a p−1 − 1) < C 2 (p/(log p) 2 ) log a.…”

On the existence of S-Diophantine quadruples

“…One can see Lemma 7 as a lower bound for a very special linear form of two q-adic logarithms (cf. Yamada's work [21] on upper bounds for v p (x p−1 − 1)). However, we will also use lower bounds for linear forms in complex logarithms.…”

“…But the problem is that to the authors knowledge for fixed p we do not know how large u p = v p (q p−1 − 1) can get. To the authors knowledge the best known upper bound for u p is due to Yamada [21] who used the very sharp results due to Bugeaud and Laurent [3] for linear forms in two p-adic logarithms. In particular, Yamada obtained that u p ≤ 283(p − 1) log 2 log p · log 2q log p + 4.…”