Further remarks on Diophantine quintuples

• Autorzy: Mihai Cipu
• Języki publikacji: EN

Abstrakty

EN

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 $d < 1.55·10^{72}$ and $b < 6.21·10^{35}$ 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.

Kategorie tematyczne

• 11B37: Recurrences
• 11D45: Counting solutions of Diophantine equations
• 11D09: Quadratic and bilinear equations
• 11J86: Linear forms in logarithms; Baker's method

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2015
• Tom: 168
• Numer: 3
• Strony: 201-219

Daty

wydano

2015

Twórcy

autor

Mihai Cipu

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

Identyfikatory

DOI

10.4064/aa168-3-1