- Department of Mathematics
University of Zagreb
Bijenicka 30
10000 Zagreb
Croatia - +385 1 460 5780
Andrej Dujella
University of Zagreb, Department of Mathematics, Faculty Member
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In this paper, we prove that if {a1, b, c, d} and {a2, b, c, d} are Diophantine quadruples with a1 < a2 < b < c < d, then a2 > 24^3 , a2 > max{36a1^3 , 300a1^2}, b < a2^{3/2}, and 16a1^2... more
In this paper, we prove that if {a1, b, c, d} and {a2, b, c, d} are Diophantine quadruples with a1 < a2 < b < c < d, then a2 > 24^3 , a2 > max{36a1^3 , 300a1^2}, b < a2^{3/2}, and 16a1^2 b^3 < c < 16a2b^3. The last inequalities imply that for a fixed Diophantine triple {b, c, d} the number of Diophantine quadruples {a, b, c, d} with a < min{b, c, d} is at most two. Moreover, we show that there are only finitely many quintuples {a1, a2, b, c, d} as above.
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The D(− 1)-quadruple conjecture states that there does not exist a set of four positive integers such that the product of any two distinct elements is one greater than a perfect square. An effective proof is given by showing that if {;a,... more
The D(− 1)-quadruple conjecture states that there does not exist a set of four positive integers such that the product of any two distinct elements is one greater than a perfect square. An effective proof is given by showing that if {;a, b, c, d}; is such a set then max{;a, b, c, d}; < 10^10^23 , leaving open a completely determined (but currently computationally infeasible large) set of exceptions to check.
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In this paper, we present the construction of an elliptic curve over Q(u) with torsion group Z/4Z and rank 6. Previously only rank 5 examples for such curves were known.
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In this survey paper, we describe several methods for constructing elliptic curves with a given torsion group and high rank over the rationals and quadratic fields. We also discuss potential applications of such curves in the elliptic... more
In this survey paper, we describe several methods for constructing elliptic curves with a given torsion group and high rank over the rationals and quadratic fields. We also discuss potential applications of such curves in the elliptic curve factorization method and their role in the construction of rational Diophantine sextuples.
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Let D be a positive integer which is not a perfect square. We consider Diophantine quintuples in the ring Z[ √ D] of the form {e, a ± b √ D, c ± d √ D} where a, b, c, d, e are integers. In this paper, we show that there exists a... more
Let D be a positive integer which is not a perfect square. We consider Diophantine quintuples in the ring Z[ √ D] of the form {e, a ± b √ D, c ± d √ D} where a, b, c, d, e are integers. In this paper, we show that there exists a Diophantine quintuple of that form for certain values of D, including D = 1 + n^2 (n + 1)^2 and some other polynomials of degree 4, and we represent its elements also as polynomials in n.
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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 ≤ i < j ≤ m, is called a Diophantine m- tuple with the property D(n) or simply D(n)-... more
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 ≤ i < j ≤ m, is called a Diophantine m- tuple with the property D(n) or simply D(n)- set. Such sets have been studied since the ancient times. In this article, we give an overview of the results from the literature about D(n)-sets and summarize our recent findings about triples of integers which are D(n)-sets for several n's. Furthermore, we include some new observations and remarks about the ways to construct such triples.
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Let E be an elliptic curve over Q given by y^2 = f(x) where f(x) = x(x − 1)(x − lambda). In this paper, we describe a construction of twists E_{; ; ; ; ; ; g(u)}; ; ; ; ; ; of rank 2 over Q(u), where g(u) are polynomials over Q. The... more
Let E be an elliptic curve over Q given by y^2 = f(x) where f(x) = x(x − 1)(x − lambda). In this paper, we describe a construction of twists E_{; ; ; ; ; ; g(u)}; ; ; ; ; ; of rank 2 over Q(u), where g(u) are polynomials over Q. The construction leads to two sets of twists: the first consists of five twists obtained by Rubin and Silverberg with a different method, while the second consists of five new twists.
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In this paper, we present details of seven elliptic curves over Q(u) with rank 2 and torsion group Z/8Z and five curves over Q(u) with rank 2 and torsion group Z/2Z× Z/6Z. We also exhibit some particular examples of curves with high rank... more
In this paper, we present details of seven elliptic curves over Q(u) with rank 2 and torsion group Z/8Z and five curves over Q(u) with rank 2 and torsion group Z/2Z× Z/6Z. We also exhibit some particular examples of curves with high rank over Q by specialization of the parameter. We present several sets of infinitely many elliptic curves in both torsion groups and rank at least 3 parametrized by elliptic curves having positive rank. In some of these sets we have performed calculations about the distribution of the root number. This has relation with recent heuristics concerning the rank bound for elliptic curves by Park, Poonen, Voight and Wood.
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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... more
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.
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.
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Number theory is a branch of mathematics that is primarily focused on the study of positive integers, or natural numbers, and their properties such as divisibility, prime factorization, or solvability of equations in integers. Number... more
Number theory is a branch of mathematics that is primarily focused on the study of positive integers, or natural numbers, and their properties such as divisibility, prime factorization, or solvability of equations in integers. Number theory has a very long and diverse history, and some of the greatest mathematicians of all time, such as Euclid, Euler and Gauss, have made significant contributions to it. Throughout its long history, number theory has often been considered as the "purest" branch of mathematics in the sense that it was the furthest from any concrete application. However, a significant change took place in the mid-1970s, and nowadays, number theory is one of the most important branches of mathematics for applications in cryptography and secure information exchange. This book is based on teaching materials from the courses Number Theory and Elementary Number Theory, which are taught at the undergraduate level studies at the Department of Mathematics, Faculty of Science, University of Zagreb, and the courses Diophantine Equations and Diophantine Approximations and Applications, which were taught at the doctoral program of mathematics at that faculty. The book thoroughly covers the content of these courses, but it also contains other related topics such as elliptic curves, which are the subject of the last two chapters in the book. The book also provides an insight into subjects that were and are at the centre of research interest of the author of the book and other members of the Croatian group in number theory, gathered around the Seminar on Number Theory and Algebra. This book is primarily intended for students of mathematics and related faculties who attend courses in number theory and its applications. However, it can also be useful to advanced high school students who are preparing for mathematics competitions in which at all levels, from the school level to international competitions, number theory has a significant role, and for doctoral students and scientists in the fields of number theory, algebra and cryptography. In the English edition, there are only minor changes compared with the Croatian version. Several misprints noticed by the author and the readers were corrected. Some information and references were updated, in particular, those related to elliptic curves rank records and new constructions of families of rational Diophantine sextuples. At just a few places in the Croatian version of the book only the references to literature in Croatian were given ; these references were expanded in the English edition with the appropriate recommendations of literature in English. The list of references has been expanded to include some recent books and papers, as well as some references which were mentioned in the text of the Croatian edition but were not included in the list of references.