Further remarks on Diophantine quintuples

Mihai Cipu¹

¹ Simion Stoilow Institute of Mathematics of the Romanian Academy Research unit no. 5 P.O. Box 1-764 RO-014700 Bucureşti, Romania

Acta Arithmetica, Tome 168 (2015) no. 3, pp. 201-219.

Résumé

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 $a b c d e$ satisfies $d 1.55 \cdot 10^{72}$ and $b 6.21 \cdot 10^{35}$ when $4 a b$, while for $b 4 a$ one has either $c=a+b +2\sqrt {ab+1}$ and $d 1.96\cdot 10^{53}$ or $c=(4ab+2)(a+b-2\sqrt {ab+1} )+2a+2b$ and $d 1.22\cdot 10^{47}$. In any case, $d 9.5\cdot b^4$.

Détail

DOI : 10.4064/aa168-3-1

Keywords: set positive integers property product predecessor perfect square called diophantine m tuple much work has done attempting prove there exist diophantine quintuples paper stringent conditions should met putative diophantine quintuple among others diophantine quintuple d satisfies cdot cdot while has either sqrt cdot b sqrt cdot cdot