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An Absolute Bound for the Size of Diophantine m-Tuples

✍ Scribed by Andrej Dujella

Publisher: Elsevier Science
Year: 2001
Tongue: English
Weight: 178 KB
Volume: 89
Category: Article
ISSN: 0022-314X
DOI: 10.1006/jnth.2000.2627

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✦ Synopsis

A set of m positive integers is called a Diophantine m-tuple if the product of its any two distinct elements increased by 1 is a perfect square. We prove that if [a, b, c] is a Diophantine triple such that b>4a and c>max[b 13 , 10 20 ] or c>max[b 5 , 10 1029 ], then there is unique positive integer d such that d>c and [a, b, c, d ] is a Diophantine quadruple. Furthermore, we prove that there does not exist a Diophantine 9-tuple and that there are only finitely many Diophantine 8-tuples.

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