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Further remarks on Diophantine quintuples

Mihai Cipu (2015)

Acta Arithmetica

A set of m positive integers with the property that the product of any two of them is the predecessor of a perfect square is called a Diophantine m-tuple. Much work has been done attempting to prove that there exist no Diophantine quintuples. In this paper we give stringent conditions that should be met by a putative Diophantine quintuple. Among others, we show that any Diophantine quintuple a,b,c,d,e with a < b < c < d < e s a t i s f i e s d < 1.55·1072 a n d b < 6.21·1035 w h e n 4 a < b , w h i l e f o r b < 4 a o n e h a s e i t h e r c = a + b + 2√(ab+1) and d < 1 . 96 · 10 53 ...

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