There is no Diophantine Quintuple

Abstract

This paper will consider the works of He, Togbé and Ziegler along with Fujita to provide an expository explanation of the proof that Diophantine quintuples do not exist. The proof of this is an extension of Dujella’s work on proving that there is no Diophantine sextuple and that there are at most finitely many Diophantine quintuples.