On Diophantine square tuples

Gyarmati, Katalin (2025) On Diophantine square tuples. FIBONACCI QUARTERLY, 63 (3). pp. 554-569. ISSN 0015-0517

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Abstract

Diophantus' problem is generalized by estimating the cardinality of a set of non-zero squares, in which the difference between any two squares is also a square. Such a set with $m$ elements is called a Diophantine square $m$-tuple. It is proved that there are infinitely many Diophantine square triples. Special cases of this problem are also studied, such as the fact that there is no Diophantine square triple containing only squares of Fibonacci numbers. For a set of integers $\mathcal A$, a non-trivial upper bound is given for the number of pairs $(a,a')$ for which $a-a'$ is a square of a Fibonacci number. Some problems and conjectures are also formulated.

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