Glasnik Matematicki, Vol. 50, No. 1 (2015), 25-34.

BOUNDS FOR DIOPHANTINE QUINTUPLES

Mihai Cipu and Yasutsugu Fujita

Simion Stoilow Institute of Mathematics of the Romanian Academy, Research unit nr. 5, P.O. Box 1-764, RO-014700 Bucharest, Romania
e-mail: Mihai.Cipu@imar.ro

Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan
e-mail: fujita.yasutsugu@nihon-u.ac.jp


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.

2010 Mathematics Subject Classification.   11D09, 11B37, 11J68.

Key words and phrases.   Diophantine m-tuples, Pell equations, hypergeometric method.


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DOI: 10.3336/gm.50.1.03


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