There is no Diophantine quintuple

He, Bo; Togbé, Alain; Ziegler, Volker

doi:10.1090/tran/7573

There is no Diophantine quintuple
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by Bo He, Alain Togbé and Volker Ziegler PDF

Trans. Amer. Math. Soc. 371 (2019), 6665-6709 Request permission

Abstract:

A set of $m$ positive integers $\{a_1, a_2, \ldots , a_m\}$ is called a Diophantine $m$-tuple if $a_i a_j + 1$ is a perfect square for all $1 \le i < j \le m$. Dujella proved that there is no Diophantine sextuple and that there are at most finitely many Diophantine quintuples. In particular, a folklore conjecture concerning Diophantine $m$-tuples states that no Diophantine quintuple exists at all. We prove this conjecture.

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