On the size of Diophantine m-tuples

By Andrej Dujella. Mathematical Proceedings of the Cambridge Philosophical Society, 2000

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Let n be a nonzero integer. A set of m positive integers {a1, a2, …, am} is said to have the property D(n) if aiaj+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 Pn-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]].

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/15acfd1e28705dbae4e06946ab708d3d1a601db9

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In Algebra and Number Theory

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