Mathematical proceedings of the Cambridge Philosophical Society, Jul 1, 2003
Let k ≥ 3 be an integer. We study the possible existence of finite sets of positive integers such... more 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 k-th power. The Greek mathematician Diophantus observed that the rational numbers 1 16 , 33 16 , 17 4 and 105 16 have the following property: the product of any two of them increased by 1 is a square of a rational number. Later, Fermat found a set of four positive integers with the above property, namely the set {1, 3, 8, 120}. We call a Diophantine m-tuple any set of m positive integers a 1 , . . . , a m such that a i a j + 1 is a perfect square whenever 1 ≤ i < j ≤ m. It was known already to Euler that there are infinitely many Diophantine quadruples (see for instance [5, pp. 513-520]). Among the broad literature on that topic, let us mention that Baker & Davenport [3] proved that {1, 3, 8} cannot be extended to a Diophantine quintuple, a result improved by Dujella & Pethő [10], who showed that even {1, 3} cannot be extended to a Diophantine quintuple. The first absolute upper bound for the size of Diophantine m-tuples was given by the second author in , where it was proved that Diophantine 9-tuples do not exist. Very recently, he was able to considerably improve upon his result, by showing [9] that there exist no Diophantine sextuple and only finitely many Diophantine quintuples. However, the question of the existence of a Diophantine quintuple remains a challenging open problem. We refer to for further references on this topic. In the present work, we are interested in an analogous problem, namely the existence of sets {a, b, c} of positive integers such that the three numbers ab + 1, ac + 1 and bc + 1 are perfect k-th powers, for an integer k ≥ 3. Examples of such triples for k=3 and k = 4 are given, respectively, by {2, 171, 25326} and {1352, 9539880, 9768370}. To our knowledge, no example of such triple is known for k ≥ 5. In order to investigate this question, we study a slightly more general problem, recently considered by Gyarmati [12]. Let N ≥ 1 and k ≥ 3 be integers. Let A and B be subsets of {1, . . . , N } such that ab + 1 is a perfect k-th power whenever a ∈ A and b ∈ B. What can be said about the cardinalities of the sets A and B ? Let |S| denote the cardinality of a finite set S. Using elementary arguments, Gyarmati proved that min{|A|, |B|} ≤ 1 + (log log N )/ log(k -1). As a corollary of our main result, we show that, except for small values of k, we have the considerably better 2000 Mathematics Subject Classification : 11D61.
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