Diophantine m-tuples and elliptic curves (original) (raw)
Related papers
Doubly regular Diophantine quadruples
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 .
On Diophantine triples and quadruples
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.
A note on 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.
Diophantine m-tuples and Elliptic Curves - front and back matters
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.
Irregular Diophantine m-tuples and elliptic curves of high rank
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.
Contributions to the Theory of Transcendental Numbers, 1984
Diophantine arithmetic is one of the oldest branches of mathematics, the search for integer or rational solutions of algebraic equations. Pythagorean triangles are an early instance. Diophantus of Alexandria wrote the first related treatise in the fourth century; it was an area extensively studied by the great mathematicians of the seventeenth century, including Euler and Fermat. The modern approach is to treat the equations as defining geometric objects, curves, surfaces, etc. The theory of elliptic curves (or curves of genus 1, which are much used in modern cryptography) was developed extensively in the twentieth century, and has had great application to Diophantine equations. This theory is used in application to the problems studied in this thesis. This thesis studies some curves of high genus, and possible solutions in both rationals and in algebraic number fields, generalizes some old results and gives answers to some open problems in the literature. The methods involve known techniques together with some ingenious tricks. For example, the equations y 2 = x 6 + k, k = −39, −47, the two previously unsolved cases for |k| < 50, are solved using algebraic number theory and the elliptic Chabauty method. The thesis also studies the genus three quartic curves F (x 2 , y 2 , z 2) = 0 where F is a homogeneous quadratic form, and extend old results of Cassels, and Bremner. It is a very delicate matter to find such curves that have no rational points, yet which do have points in odd-degree extension fields of the rationals. The principal results of the thesis are related to surfaces where the theory is much less well known. In particular, the thesis studies some specific families of surfaces, and give a negative answer to a question in the literature regarding representation of integers n in the form n = (x+y +z +w)(1/x+1/y +1/z +1/w). Further, an example, the first such known, of a quartic surface x 4 + 7y 4 = 14z 4 + 18w 4 is given with remarkable properties: it is everywhere locally solvable, yet has no nonzero rational point, despite having a point in (non-trivial) odd-degree extension fields i of the rationals. The ideas here involve manipulation of the Hilbert symbol, together with the theory of elliptic curves. ii ACKNOWLEDGEMENTS I would like to thank my advisor Professor Andrew Bremner for his guidance, his generosity, his encouragement and his kindness during my graduate years. Without his help and support, I will not be able to finish the thesis. I show my most respect to him, both his personality and his mathematical expertise. I would like to thank Professor Susanna Fishel for some talks we had. These talks did encourage me a lot at the beginning of my graduate years. I would like to thank other members of my Phd committee, Professor John Quigg, Professor John Jones, and Professor Nancy Childress. I would like to thank the school of mathematics and statistical sciences at Arizona State University for all the funding and support. And finally, I would like to thank the members in my family. My grandmother, my father, my mom, Mr Phuong and his wife Mrs Doi and their son Phi, and to my cousin Mr Tan for all of their constant support and encouragement during my undergraduate and my graduate years.
There are only finitely many Diophantine quintuples
Journal fur die Reine und Angewandte Mathematik, 2004
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. Diophantus found a set of four positive rationals with the above property. The first Diophantine quadruple was found by Fermat (the set {1, 3, 8, 120}). Baker and Davenport proved that this particular quadruple cannot be extended to a Diophantine quintuple.
On elliptic curves induced by rational Diophantine quadruples
Proc. Japan Acad. Ser. A Math. Sci., 2022
In this paper, we consider elliptic curves induced by rational Dio-phantine quadruples, i.e. sets of four nonzero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups Z/2Z × Z/kZ for k = 2, 4, 6, 8, there are infinitely many rational Dio-phantine quadruples with the property that the induced elliptic curve has this torsion group. We also construct curves with moderately large rank in each of these four cases.
Conjectures and results on the size and number of Diophantine tuples
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.