On a problem of Diophantus for higher powers

Mathematical Proceedings of the Cambridge …, 2002

Acta Arith, 1993

… computational, and algebraic aspects: proceedings of …, 1998

A Diophantine m-tuple with the property D(n), where n is an integer, is defined as a set of m positive integers with the property that the product of its any two distinct elements increased by n is a perfect square. It is known that if n is of the form 4k + 2, then there does not exist a Diophantine quadruple with the property D(n). The author has formerly proved that if n is not of the form 4k + 2 and n ∈ {−15, −12, −7, −4, −3, −1, 3, 5, 7, 8, 12, 13, 15, 20, 21, 24, 28, 32, 48, 60, 84}, then there exist at least two distinct Diophantine quadruples with the property D(n).

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 .

Acta Mathematica Hungarica, 2009

Let A and k be positive integers. We study the Diophantine quadruples k, A 2 k + 2A, (A + 1) 2 k + 2(A + 1), d .

2007

Springer eBooks, 1998

Mathematical communications, 1997

In this paper we describe the author's results concerning the problem of the existence of a set of four or five positive integers with the property that the product of its any two distinct elements increased by a fixed integer n is a perfect square.

数理解析研究所講究録 = RIMS Kokyuroku, 2018

For a nonzero integer n, a set of distinct nonzero integers \{a_{1} : a_{2} : a_{m}\} such that a_{i}a_{j}+n is a perfect square for all 1\leq i<j\leq m_{:} is called a Diophantine m-tuple with the property D(n) or simply D(n)-set. Such sets have been studied bince the ancient times. In this article, we give an overview of the results from the literature about D(n)-betb and summarize our recent findings about triples of integers which aleD(n)-sets for several n^{:}s. Furthermore, we include some new observations and remarks about the ways to construct such triples.

Fibonacci Quart., to appear