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Abstract

A set of $m$ positive integers with the property that the product of any two of them is the predecessor of a perfect square is called a Diophantine $m$-tuple. Much work has been done attempting to prove that there exist no Diophantine quintuples. In this paper we give stringent conditions that should be met by a putative Diophantine quintuple. Among others, we show that any Diophantine quintuple $\{a,b,c,d,e\}$ with $a< b< c< d< e$ satisfies $d< 1.55 \cdot 10^{72}$ and $b < 6.21 \cdot 10^{35}$ when $4 a < b$, while for $b < 4 a$ one has either $c=a+b +2\sqrt {ab+1}$ and $d< 1.96\cdot 10^{53}$ or $c=(4ab+2)(a+b-2\sqrt {ab+1} )+2a+2b$ and $d < 1.22\cdot 10^{47}$. In any case, $d < 9.5\cdot b^4$.