Original scientific paper

https://doi.org/10.3336/gm.50.1.03

Bounds for Diophantine quintuples

Mihai Cipu ; Simion Stoilow Institute of Mathematics of the Romanian Academy, Research unit nr. 5, P.O. Box 1-764, RO-014700 Bucharest, Romania
Yasutsugu Fujita ; Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan

Abstract

A set of m positive integers {a1,...,am} is called a Diophantine m-tuple if the product of any two elements in the set increased by one is a perfect square. The conjecture according to which there does not exist a Diophantine quintuple is still open. In this paper, we show that if {a,b,c,d,e} is a Diophantine quintuple with a < b < c < d < e , then b >3a; moreover, b > max{21 a, 2 a3/2} in case c>a+b+2(ab+1)1/2.

Keywords

Diophantine m-tuples; Pell equations; hypergeometric method

Hrčak ID:

140081

URI

https://hrcak.srce.hr/140081

Publication date:

15.6.2015.

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