On the size of Diophantine <i>m</i>-tuples

Abstract

Let n be a nonzero integer. A set of m positive integers { a 1 , a 2 , …, a m } is said to have the property D ( n ) if a i a j + n is a perfect square for all 1 [les ] i [les ] j [les ] m . Such a set is called a Diophantine m -tuple (with the property D ( n )), or P n -set of size m . Diophantus found the quadruple {1, 33, 68, 105} with the property D (256). The first Diophantine quadruple with the property D (1), the set {1, 3, 8, 120}, was found by Fermat (see [ 8 , 9 ]). Baker and Davenport [ 3 ] proved that this Fermat’s set cannot be extended to the Diophantine quintuple, and a famous conjecture is that there does not exist a Diophantine quintuple with the property D (1). The theorem of Baker and Davenport has been recently generalized to several parametric families of quadruples [ 12 , 14 , 16 ], but the conjecture is still unproved. On the other hand, there are examples of Diophantine quintuples and sextuples like {1, 33, 105, 320, 18240} with the property D (256) [ 11 ] and {99, 315, 9920, 32768, 44460, 19534284} with the property D (2985984) [ 19 ]].

Locations

• Mathematical Proceedings of the Cambridge Philosophical Society - View
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