Further remarks on Diophantine quintuples

Mihai Cipu

Acta Arithmetica (2015)

  • Volume: 168, Issue: 3, page 201-219
  • ISSN: 0065-1036

Abstract

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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 or c = (4ab+2)(a+b-2√(ab+1)) + 2a + 2b and d < 1 . 22 · 10 47 . In any case, d < 9.5·b⁴.

How to cite

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Mihai Cipu. "Further remarks on Diophantine quintuples." Acta Arithmetica 168.3 (2015): 201-219. <http://eudml.org/doc/279167>.

@article{MihaiCipu2015,
abstract = {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 $d < 1.55·1072$ and $b < 6.21·1035$ when 4a < b, while for b < 4a one has either $c = a + b + 2√(ab+1) and $d < 1.96·10^\{53\}$ or c = (4ab+2)(a+b-2√(ab+1)) + 2a + 2b and $d < 1.22·10^\{47\}$. In any case, d < 9.5·b⁴.},
author = {Mihai Cipu},
journal = {Acta Arithmetica},
keywords = {Diophantine m-tuples; Pell equations; linear forms in logarithms},
language = {eng},
number = {3},
pages = {201-219},
title = {Further remarks on Diophantine quintuples},
url = {http://eudml.org/doc/279167},
volume = {168},
year = {2015},
}

TY - JOUR
AU - Mihai Cipu
TI - Further remarks on Diophantine quintuples
JO - Acta Arithmetica
PY - 2015
VL - 168
IS - 3
SP - 201
EP - 219
AB - 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 $d < 1.55·1072$ and $b < 6.21·1035$ when 4a < b, while for b < 4a one has either $c = a + b + 2√(ab+1) and $d < 1.96·10^{53}$ or c = (4ab+2)(a+b-2√(ab+1)) + 2a + 2b and $d < 1.22·10^{47}$. In any case, d < 9.5·b⁴.
LA - eng
KW - Diophantine m-tuples; Pell equations; linear forms in logarithms
UR - http://eudml.org/doc/279167
ER -

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