There are only finitely many Diophantine quintuples

1998

Diophantus noted that the rational numbers 1/16, 33/16, 17/4 and 105/16 have the following property: the product of any two of them increased by 1 is a square of a rational number.

Acta Mathematica Spalatensia, 2021

A set of m distinct nonzero rationals {a 1 , a 2 ,. .. , am} such that a i a j + 1 is a perfect square for all 1 ≤ i < j ≤ m, is called a rational Dio-phantine m-tuple. It is proved recently that there are infinitely many rational Diophantine sextuples. In this paper, we construct infinite families of rational Diophantine sextuples with special structure, namely the sextuples containing quadruples and quintuples of certain type.

International Mathematics Research Notices, 2016

A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple. In this paper, we prove that there exist infinitely many rational Diophantine sextuples.

Number Theory - Diophantine problems, uniform distribution and applications, Festschrift in honour of Robert F. Tichy's 60th birthday (C. Elsholtz, P. Grabner, Eds.), Springer-Verlag, Berlin, 2017

A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple, and Dujella, Kazalicki, Mikic and Szikszai recently proved that there exist infinitely many rational Diophantine sextuples. In this paper, generalizing the work of Piezas, we describe a method for generating new parametric formulas for rational Diophantine sextuples.

1998

A Diophantine m-tuple with the property D(n) is a set {a 1 , a 2 , . . . a m } of positive integers such that for 1 ≤ i < j ≤ m, the number a i a j + n is a perfect square. In the present paper we give necessary conditions that the elements a i of a set {a 1 , a 2 , a 3 , a 4 , a 5 } must satisfy modulo 4 in order to be a Diophantine quintuple.

Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, 2020

For a nonzero integer n, a set of m distinct nonzero integers {a 1 , a 2 ,. .. , am} such that a i a j + n is a perfect square for all 1 ≤ i < j ≤ m, is called a D(n)-m-tuple. In this paper, by using properties of so-called regular Diophantine m-tuples and certain family of elliptic curves, we show that there are infinitely many essentially different sets consisting of perfect squares which are simultaneously D(n 1)-quadruples and D(n 2)-quadruples with distinct nonzero squares n 1 and n 2 .

Glasgow Mathematical Journal, 2007

It is proven that if k ≥ 2 is an integer and d is a positive integer such that the product of any two distinct elements of the set {k -1, k + 1, 16k 3 -4k, d} increased by 1 is a perfect square, then d = 4k or d = 64k 5 -48k 3 + 8k. Together with a recent result of Fujita, this shows that all Diophantine quadruples of the form {k -1, k + 1, c, d} are regular. 2000 Mathematics Subject Classification. 11D09, 11D25, 11J86, 11Y50. A Diophantine m-tuple is a set of m positive integers such that the product of any two of them increased by 1 gives a perfect square. Diophantus himself studied sets of positive rationals with the same property, while the first Diophantine quadruple, namely the set {1, 3, 8, 120}, was found by Fermat ([4, 5, 13]). In 1969, Baker and Davenport [1] proved that the Fermat set cannot be extended to a Diophantine quintuple. There are several generalizations of the result of Baker and Davenport. In 1997, Dujella [6] proved that the Diophantine triples of the form {k -1, k + 1, 4k}, for k ≥ 2, cannot be extended to a Diophantine quintuple (k = 2 gives the Baker-Davenport result), while in 1998, Dujella and Peth ö [9] proved that the Diophantine pair {1, 3} cannot be extended to a Diophantine quintuple. Recently, Fujita [12] obtained a result which is common generalization of the results from [6] and . Namely, he proved that the Diophantine pairs of the form {k -1, k + 1}, for k ≥ 2 cannot be extended to a Diophantine quintuple. A folklore conjecture is that there does not exist a Diophantine quintuple. An important progress towards its resolution was done in 2004 by Dujella [8], who proved that there are only finitely many Diophantine quintuples. The stronger version of this conjecture states that if {a, b, c, d} is a Diophantine quadruple and d > max{a, b, c}, then d = a + b + c + 2abc + 2 (ab + 1)(ac + 1)(bc + 1). Diophantine quadruples of .

Mathematical Proceedings of the Cambridge …, 2002

Journal of Number Theory, 2001

A set of m positive integers is called a Diophantine m-tuple if the product of its any two distinct elements increased by 1 is a perfect square. We prove that if {a, b, c} is a Diophantine triple such that b > 4a and c > max{b 13 , 10 20 } or c > max{b 5 , 10 1029 }, then there is unique positive integer d such that d > c and {a, b, c, d} is a Diophantine quadruple. Furthermore, we prove that there does not exist a Diophantine 9-tuple and that there are only finitely many Diophantine 8-tuples.

Notes on Number Theory and Discrete Mathematics, 2015

In this paper we consider Diophantine triples {a, b, c}, (denoted D(n)-3-tuples) and give necessary and sufficient conditions for existence of integer n given the 3-tuple {a, b, c}, so that ab + n, ac + n, bc + n are all squares of integers. Several examples as applications of the main results, related to both Diophantine triples and quadruples, are given.