Glasnik Matematicki, Vol. 45, No.1 (2010), 15-29.

THE NUMBER OF DIOPHANTINE QUINTUPLES

Yasutsugu Fujita

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 {a1, ... ,am} of m distinct positive integers is called a Diophantine m-tuple if aiaj+1 is a perfect square for all i, j with 1 ≤ i < jm. It is known that there does not exist a Diophantine sextuple and that there exist only finitely many Diophantine quintuples. In this paper, we first show that for a fixed Diophantine triple {a,b,c} with a < b < c, the number of Diophantine quintuples {a,b,c,d,e} with c < d < e is at most four. Using this result, we further show that the number of Diophantine quintuples is less than 10276, which improves the bound 101930 due to Dujella.

2000 Mathematics Subject Classification.   11D09, 11J68, 11J86.

Key words and phrases.   Simultaneous Diophantine equations, Diophantine tuples.


Full text (PDF) (free access)

DOI: 10.3336/gm.45.1.02


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