Glasnik matematički, Vol. 45 No. 1, 2010.
Original scientific paper
https://doi.org/10.3336/gm.45.1.02
The number of Diophantine quintuples
Yasutsugu Fujita
; Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan
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 < j ≤ m. 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.
Keywords
Simultaneous Diophantine equations; Diophantine tuples
Hrčak ID:
52364
URI
Publication date:
17.5.2010.
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