On an $S$-unit variant of Diophantine $m$-tuples

Abstract

Let S be a fixed set of primes and let a 1 , . . . , a m be positive distinct integers. We call the m-tuple (a 1 , . . . , a m ) S-Diophantine, if for all i = j the integers aiaj + 1 = si,j are S-integers. In this paper we show that if |S| = 2, then under some technical restrictions no S-Diophantine quadruple exists.

30 years of collaboration

“…More results can be found for example in [62,63,64,18]. Other lines of related research which we will not go into can e.g.…”

S-Diophantine Quadruples With Two Primes Congruent to 3 Modulo 4

“…We start with a very useful lemma (see [7,Lemma 2]) which excludes some divisibility relations for S-Diophantine triples. This lemma is exactly Lemma 2 in [7].…”

“…Later Fuchs, Luca and the first author [5] replaced ⇤ by terms of a given binary recurrence sequence, in particular, the Fibonacci sequence [6]. Recently the authors substituted ⇤ by S-units [7]. For complete overview we suggest Dujella's web page on Diophantine tuples [2].…”

“…In a recent paper [7] the authors showed that if S = {p, q} and C(⇠) < p < q < p ⇠ for some ⇠ > 1 and for some explicitly computable constant C(⇠), then no S-Diophantine quadruple exists. This result and numerical experiments (see [7, Lemma 9], where we found no quadruples with 1  a < b < c < d  1000) raise the question of whether or not an S-Diophantine quadruple with |S| = 2 exists.…”

DIO 3 - Tuples for Special Numbers - I

“…Many generalizations of this problem (1) were considered since antiquity, for example by adding a fixed integer n instead of 1, looking kth powers instead of squares or considering the powers over domains other than Z or Q. Many mathematicians consider the problem of the existence of Diophantine quadruples with the property D(n) for any arbitrary integer n and also for any linear polynomials in n. In this context one may refer [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. The above results motivated us the following definition:…”