There are no Diophantine Quintuples

Volker Ziegler (Paris Lodron-University Salzburg)

Frank Adams 1,

A \(m\)-tuple of distinct positive integers \((a_1,\dots,a_m)\) is called a Diophantine \(m\)-tuple if \(a_ia_j+1\) is a perfect square for all \(i\neq j\). It was a long outstanding question whether a Diophantine quintuple exists. In a recent paper joint with Bo He and
Alain Togbè we proved that no Diophantine quintuple exists. After a short introduction to the problem we present the new ideas that led to the proof of the so-called Diophantine quintuple conjecture.
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