On a problem of Diophantus for higher powers

Abstract

Let k 3 be an integer. We study the possible existence of finite sets of positive integers such that the product of any two of them increased by 1 is a kth power.

Fibonacci Diophantine triples

“…In the rational case, it is not known that the size m of the Diophantine m-tuples must be bounded and a few examples with m = 6 are known by the work of Gibbs [8]. We also note that some generalization of this problem for squares replaced by higher powers (of fixed, or variable exponents) were treated by many authors (see [1,2,9,13] and [10]). …”

On the size of sets whose elements have perfect power $n$-shifted products

“…Further, we prove that assuming the abc-conjecture we already have |A| < C(n), where C(n) is a constant depending only on n. In view of our construction in the proof of Theorem 2, the dependence of C(n) on n is necessary. To prove this result we extend a theorem of Bugeaud and Dujella [3] concerning shifted products which are k-th powers (Theorem 3). Assuming the abc-conjecture we obtain a bound in terms of n for all but one a i , provided that the exponents k ij in a i a j + n = x kij ij are sufficiently large (Lemma 1).…”

“…As an alternative, but also natural generalization of Diophantine m-tuples, Bugeaud and Dujella [3] considered sets A of positive integers with the property that ab + 1 = x k ab whenever a, b are distinct elements of A and k is an integer with k ≥ 2. Such sets are called k-th power Diophantine tuples.…”

“…In [3,Corollary 4] absolute upper bounds for the numbers E k , k ≥ 3 were obtained. More precisely, it was proved that E 3 ≤ 7, E 4 ≤ 5, E 5 ≤ 5, E k ≤ 4 for 6 ≤ k ≤ 176, and E k ≤ 3 for k ≥ 177.…”

Lucas Diophantine Triples

“…A few examples with m = 6 are known by the work of Gibbs [7]. Several generalizations of this problem, when the squares are replaced by higher powers of fixed, or variable exponents, were treated in many papers (see [1], [2], [8], [9]) and [10]). …”