There is no Diophantine quintuple

You need to know: Perfect square

Background: A set of $m$ distinct positive integers $\{a_1, a_2, \dots, a_m\}$ is called a Diophantine $m$ -tuple if $a_ia_j+1$ is a perfect square for all $1\leq i<j\leq m$ . In particular, it is called Diophantine quadruple, quintuple, and sextuple for $m=4$ , $m=5$ , and $m=6$ , respectively.

The Theorem: On 13th October 2016, Bo He, Alain Togbè, and Volker Ziegler submitted to arxiv a paper in which they proved that there is no Diophantine quintuple.

Short context: More than two thousands years ago, Diophantus noticed that the set of rational numbers $\{\frac{1}{16}, \frac{33}{16}, \frac{17}{4}, \frac{105}{16}\}$ has the property that the product of any two of them plus one is a square of a rational number. Later, Fermat found positive integers $\{1,3,8,120\}$ with this property, and Euler proved that there are infinitely many such quadruples. A long-standing folklore conjecture predicts that no five positive integers with this property exist. As a partial progress, Dujella proved in 2004 that there is no Diophantine sextuple and that there can be at most finitely many Diophantine quintuples. The Theorem confirms the conjecture in full.