Diophantine m-tuples and elliptic curves

Developments in Mathematics, 2024

This book provides an overview of the main results and problems concerning Diophantine m-tuples, i.e., sets of integers or rationals with the property that the product of any two of them is one less than a square, and their connections with elliptic curves. It presents the contributions of famous mathematicians of the past, like Diophantus, Fermat and Euler, as well as some recent results of the author and his collaborators. The book presents fragments of the history of Diophantine m-tuples, emphasising the connections between Diophantine m-tuples and elliptic curves. It shows how elliptic curves are used to solve some longstanding problems on Diophantine m-tuples, such as the existence of infinite families of rational Diophantine sextuples. On the other hand, rational Diophantine m-tuples are used to construct elliptic curves with interesting Mordell–Weil groups, including curves of record rank with a given torsion group. The book contains concrete algorithms and advice on how to use the software package PARI/GP for solving computational problems relevant to the book's topics. This book is primarily intended for researchers and graduate students in Diophantine equations and elliptic curves. However, it can be of interest to other mathematicians interested in number theory and arithmetic geometry. The prerequisites are on the level of a standard first course in elementary number theory. Background in elliptic curves, Diophantine equations and Diophantine approximations is provided in the book. An interested reader may consult also the recent Number Theory book by the author. The author gave a course based on the preliminary version of this book in the academic year 2021/2022 for PhD students at the University of Zagreb. On the course web page, additional materials, like homework exercises (mostly included in the book in the exercise sections at the end of each chapter), seminar topics and links to relevant software, can be found. The book could be used as a textbook for a specialized graduate course, and it may also be suitable for a second reading supplement reference in any course on Diophantine equations and/or elliptic curves at the graduate or undergraduate level.

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.

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 .

Mathematical Proceedings of the Cambridge …, 2002

PROCEEDINGS-JAPAN ACADEMY SERIES A …, 2000

A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them is one less than a perfect square. In this paper we characterize the notions of regular Diophantine quadruples and quintuples, introduced by Gibbs, by means of elliptic curves. Motivated by these characterizations, we find examples of elliptic curves over Q with torsion group Z/2Z × Z/2Z and with rank equal 8.

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.

2008

The problem of the construction of Diophantine m-tuples, i.e. sets with the property that the product of any two of its distinct elements is one less then a square, has a very long history. In this survey, we describe several conjectures and recent results concerning Diophantine m-tuples and their generalizations.

The problem of the construction of Diophantine m-tuples, i.e. sets with the property that the product of any two of its distinct elements is one less then a square, has a very long history. In this survey, we describe several conjectures and recent results concerning Diophantine m-tuples and their generalizations.

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 Philosophical Society, 2002

Let n be a nonzero integer and assume that a set S of positive integers has the property that xy + n is a perfect square whenever x and y are distinct elements of S. In this paper we find some upper bounds for the size of the set S. We prove that if |n| ≤ 400 then |S| ≤ 32, and if |n| > 400 then |S| < 267.81 log |n| (log log |n|) 2 . The question whether there exists an absolute bound (independent on n) for |S| still remains open.