Further remarks on Diophantine quintuples

Mihai Cipu

Mihai Cipu

Acta Arithmetica, Tome 168 (2015), p. 201-219 / Harvested from The Polish Digital Mathematics Library

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 $s a t i s f i e s$ 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 $d < 1.96 \cdot 10^{53}$ or c = (4ab+2)(a+b-2√(ab+1)) + 2a + 2b and $d < 1.22 \cdot 10^{47}$ . In any case, d < 9.5·b⁴.

Publié le : 2015-01-01

Zbl 06430567

EUDML-ID : urn:eudml:doc:279167

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     author = {Mihai Cipu},
     title = {Further remarks on Diophantine quintuples},
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     volume = {168},
     year = {2015},
     pages = {201-219},
     zbl = {06430567},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa168-3-1}
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Mihai Cipu. Further remarks on Diophantine quintuples. Acta Arithmetica, Tome 168 (2015) pp. 201-219. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa168-3-1/