Bo He, Alain Togbè, Volker Ziegler
Published 2016-10-13Version 1
A set of $m$ positive integers $\{a_1, a_2, \dots , a_m\}$ is called a Diophantine $m$-tuple if $a_i a_j + 1$ is a perfect square for all $1 \le i < j \le m$. In 2004 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. In this paper we prove this conjecture.
Non-extensibility of the pair $\{1, 3\}$ to a Diophantine quintuple in $\mathbb{Z}\left[\sqrt{-d}\right]$
Perfect squares have at most five divisors close to its square root
Difference of powers of consecutive primes which are perfect squares
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