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Andrej Dujella

University of Zagreb, Department of Mathematics, Faculty Member

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Phone: +385 1 460 5780
Address: Department of Mathematics
University of Zagreb
Bijenicka 30
10000 Zagreb
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Papers by Andrej Dujella

Asymptotics of D(q)-pairs and triples via L-functions of Dirichlet characters

Ramanujan J., 2025

Let q be a non-zero integer. A D(q)-m-tuple is a set of m distinct positive integers {a1, a2, . .... more Let q be a non-zero integer. A D(q)-m-tuple is a set of m distinct positive integers
{a1, a2, . . . , am} such that aiaj + q is a perfect square for all 1 ⩽ i < j ⩽ m. By counting
integer solutions x ∈ [1, b] of congruences x2 ≡ q (mod b) with b ⩽ N , we count D(q)-pairs with
both elements up to N, and give estimates on asymptotic behaviour. We show that for prime
q, the number of such D(q)-pairs and D(q)-triples grows linearly with N . Up to a factor of 2,
the slope of this linear function is the quotient of the value of the L-function of an appropriate
Dirichlet character (usually a Kronecker symbol) and of ζ(2).

On the indecomposability of polynomials

Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, 2005

Applying a combinatorial lemma a new sufficient condition for the indecomposability of integer po... more

Some Diophantine properties of the sequence of S -units

Journal of Number Theory, May 1, 2014

We prove some diophantine properties of the sequence of S-units.

A note on Diophantine quintuples

De Gruyter eBooks, Mar 24, 2012

ON THE FAMILY OF DIOPHANTINE TRIPLES {<i>k</i>− 1,<i>k</i>+ 1, 16<i>k</i><sup>3</sup>− 4<i>k</i>}

Glasgow Mathematical Journal, May 1, 2007

It is proven that if k ≥ 2 is an integer and d is a positive integer such that the product of any... more 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 .

The problem of Diophantus and Davenport

Mathematical Communications, Dec 20, 1997

In this paper we describe the author's results concerning the problem of the existence of a set o... more

An example of elliptic curve over Q with rank equal to 15

Proc. Japan Acad. Ser. A Math. Sci., Sep 1, 2002

We construct an elliptic curve over Q with non-trivial 2-torsion point and rank exactly equal to 15.

On the size of Diophantine <i>m</i>-tuples

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 ... more 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.

Products of three factorial

Publicationes Mathematicae Debrecen, Jul 1, 2014

We study a question of Erdős and Graham on products of three factorials being a square.

$Research paper thumbnail of On the rank of elliptic curves over $\mathbf{Q}(i)$ with torsion group $\mathbf{Z}/4\mathbf{Z} \times \mathbf{Z}/4\mathbf{Z}$$

On the rank of elliptic curves over $\mathbf{Q}(i)$ with torsion group $\mathbf{Z}/4\mathbf{Z} \times \mathbf{Z}/4\mathbf{Z}$

Proceedings of the Japan Academy. Series A, Mathematical sciences, Jun 1, 2010

Ranks of Elliptic Curves with Prescribed Torsion over Number Fields

International Mathematics Research Notices, Feb 15, 2013

We study the structure of the Mordell-Weil group of elliptic curves over number fields of degree ... more We study the structure of the Mordell-Weil group of elliptic curves over number fields of degree 2, 3, and 4. We show that if T is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup T is empty, or it contains curves of rank 0 as well as curves of positive rank. We prove a similar but slightly weaker result for cubic and quartic fields. On the other hand, we find a group T and a quartic field K such that among the elliptic curves over K with torsion subgroup T , there are curves of positive rank, but none of rank 0. We find examples of elliptic curves with positive rank and given torsion in many previously unknown cases. We also prove that all elliptic curves over quadratic fields with a point of order 13 or 18 and all elliptic curves over quartic fields with a point of order 22 are isogenous to one of their Galois conjugates and, by a phenomenon that we call false complex multiplication, have even rank. Finally, we discuss connections with elliptic curves over finite fields and applications to integer factorization.

Bounds for the size of sets with the property D(n)

Glasnik Matematicki, Dec 15, 2004

Let n be a nonzero integer and a 1 < a 2 < • • • < am positive integers such that a i a j +n is a... more

A family of quartic Thue inequalities

Acta Arithmetica, 2004

$Research paper thumbnail of An example of elliptic curve over $\mathbf {Q}$ with rank equal to 15$

An example of elliptic curve over $\mathbf {Q}$ with rank equal to 15

Proceedings of the Japan Academy. Series A, Mathematical sciences, Sep 1, 2002

We construct an elliptic curve over Q with non-trivial 2-torsion point and rank exactly equal to 15.

On the size of sets whose elements have perfect power $n$-shifted products

Publicationes Mathematicae Debrecen, Dec 1, 2011

On the size of sets whose elements have perfect power n-shifted products By ATTILA B ÉRCZES (Debr... more

On a Variation of a Congruence of Subbarao

Journal of The Australian Mathematical Society, Oct 1, 2012

We study positive integers n such that nφ(n) ≡ 2 mod σ(n), where φ(n) and σ(n) are the Euler func... more We study positive integers n such that nφ(n) ≡ 2 mod σ(n), where φ(n) and σ(n) are the Euler function and the sum of divisors function of the positive integer n, respectively. We give a general ineffective result showing that there are only finitely many such n whose prime factors belong to a fixed finite set. When this finite set consists only of the two primes 2 and 3 we use continued fractions to find all such positive integers n.

On power values of pyramidal numbers, II

by Andrej Dujella and Kalman Gyory

Acta Arith., Dec 7, 2023

For m ≥ 3, we define the mth order pyramidal number by In a previous paper, written by the first-... more For m ≥ 3, we define the mth order pyramidal number by In a previous paper, written by the first-, second-, and fourth-named authors, all solutions to the equation Pyr m (x) = y 2 are found in positive integers x and y, for 6 ≤ m ≤ 100. In this paper, we consider the question of higher powers, and find all solutions to the equation Pyr m (x) = y n in positive integers x, y, and n, with n ≥ 3, and 5 ≤ m ≤ 50. We reduce the problem to a study of systems of binomial Thue equations, and use a combination of local arguments, the modular method via Frey curves, and bounds arising from linear forms in logarithms to solve the problem.

D(n)-quintuples with square elements

arXiv (Cornell University), Nov 3, 2020

For an integer n, a set of m distinct nonzero integers {a 1 , a 2 , . . . , am} such that a i a j... more For an 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, we show that there are infinitely many essentially different D(n)-quintuples with square elements. We obtained this result by constructing genus one curves on a certain double cover of A 2 branched along four curves.

On fundamental units of real quadratic fields of class number 1

Archiv der Mathematik, Jun 29, 2019

In this paper, we give a nontrivial lower bound for the fundamental unit of norm -1 of a real qua... more

On the size of Diophantine m-tuples

arXiv (Cornell University), Feb 11, 2000

Let n be a nonzero integer and assume that a set S of positive integers has the property that xy ... more 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.

Asymptotics of D(q)-pairs and triples via L-functions of Dirichlet characters

Ramanujan J., 2025

Let q be a non-zero integer. A D(q)-m-tuple is a set of m distinct positive integers {a1, a2, . .... more Let q be a non-zero integer. A D(q)-m-tuple is a set of m distinct positive integers
{a1, a2, . . . , am} such that aiaj + q is a perfect square for all 1 ⩽ i < j ⩽ m. By counting
integer solutions x ∈ [1, b] of congruences x2 ≡ q (mod b) with b ⩽ N , we count D(q)-pairs with
both elements up to N, and give estimates on asymptotic behaviour. We show that for prime
q, the number of such D(q)-pairs and D(q)-triples grows linearly with N . Up to a factor of 2,
the slope of this linear function is the quotient of the value of the L-function of an appropriate
Dirichlet character (usually a Kronecker symbol) and of ζ(2).

On the indecomposability of polynomials

Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, 2005

Applying a combinatorial lemma a new sufficient condition for the indecomposability of integer po... more

Some Diophantine properties of the sequence of S -units

Journal of Number Theory, May 1, 2014

We prove some diophantine properties of the sequence of S-units.

A note on Diophantine quintuples

De Gruyter eBooks, Mar 24, 2012

ON THE FAMILY OF DIOPHANTINE TRIPLES {<i>k</i>− 1,<i>k</i>+ 1, 16<i>k</i><sup>3</sup>− 4<i>k</i>}

Glasgow Mathematical Journal, May 1, 2007

It is proven that if k ≥ 2 is an integer and d is a positive integer such that the product of any... more 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 .

The problem of Diophantus and Davenport

Mathematical Communications, Dec 20, 1997

In this paper we describe the author's results concerning the problem of the existence of a set o... more

An example of elliptic curve over Q with rank equal to 15

Proc. Japan Acad. Ser. A Math. Sci., Sep 1, 2002

We construct an elliptic curve over Q with non-trivial 2-torsion point and rank exactly equal to 15.

On the size of Diophantine <i>m</i>-tuples

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 ... more 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.

Products of three factorial

Publicationes Mathematicae Debrecen, Jul 1, 2014

We study a question of Erdős and Graham on products of three factorials being a square.

$Research paper thumbnail of On the rank of elliptic curves over $\mathbf{Q}(i)$ with torsion group $\mathbf{Z}/4\mathbf{Z} \times \mathbf{Z}/4\mathbf{Z}$$

On the rank of elliptic curves over $\mathbf{Q}(i)$ with torsion group $\mathbf{Z}/4\mathbf{Z} \times \mathbf{Z}/4\mathbf{Z}$

Proceedings of the Japan Academy. Series A, Mathematical sciences, Jun 1, 2010

Ranks of Elliptic Curves with Prescribed Torsion over Number Fields

International Mathematics Research Notices, Feb 15, 2013

We study the structure of the Mordell-Weil group of elliptic curves over number fields of degree ... more We study the structure of the Mordell-Weil group of elliptic curves over number fields of degree 2, 3, and 4. We show that if T is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup T is empty, or it contains curves of rank 0 as well as curves of positive rank. We prove a similar but slightly weaker result for cubic and quartic fields. On the other hand, we find a group T and a quartic field K such that among the elliptic curves over K with torsion subgroup T , there are curves of positive rank, but none of rank 0. We find examples of elliptic curves with positive rank and given torsion in many previously unknown cases. We also prove that all elliptic curves over quadratic fields with a point of order 13 or 18 and all elliptic curves over quartic fields with a point of order 22 are isogenous to one of their Galois conjugates and, by a phenomenon that we call false complex multiplication, have even rank. Finally, we discuss connections with elliptic curves over finite fields and applications to integer factorization.

Bounds for the size of sets with the property D(n)

Glasnik Matematicki, Dec 15, 2004

Let n be a nonzero integer and a 1 < a 2 < • • • < am positive integers such that a i a j +n is a... more

A family of quartic Thue inequalities

Acta Arithmetica, 2004

$Research paper thumbnail of An example of elliptic curve over $\mathbf {Q}$ with rank equal to 15$

An example of elliptic curve over $\mathbf {Q}$ with rank equal to 15

Proceedings of the Japan Academy. Series A, Mathematical sciences, Sep 1, 2002

We construct an elliptic curve over Q with non-trivial 2-torsion point and rank exactly equal to 15.

On the size of sets whose elements have perfect power $n$-shifted products

Publicationes Mathematicae Debrecen, Dec 1, 2011

On the size of sets whose elements have perfect power n-shifted products By ATTILA B ÉRCZES (Debr... more

On a Variation of a Congruence of Subbarao

Journal of The Australian Mathematical Society, Oct 1, 2012

We study positive integers n such that nφ(n) ≡ 2 mod σ(n), where φ(n) and σ(n) are the Euler func... more We study positive integers n such that nφ(n) ≡ 2 mod σ(n), where φ(n) and σ(n) are the Euler function and the sum of divisors function of the positive integer n, respectively. We give a general ineffective result showing that there are only finitely many such n whose prime factors belong to a fixed finite set. When this finite set consists only of the two primes 2 and 3 we use continued fractions to find all such positive integers n.

On power values of pyramidal numbers, II

by Andrej Dujella and Kalman Gyory

Acta Arith., Dec 7, 2023

For m ≥ 3, we define the mth order pyramidal number by In a previous paper, written by the first-... more For m ≥ 3, we define the mth order pyramidal number by In a previous paper, written by the first-, second-, and fourth-named authors, all solutions to the equation Pyr m (x) = y 2 are found in positive integers x and y, for 6 ≤ m ≤ 100. In this paper, we consider the question of higher powers, and find all solutions to the equation Pyr m (x) = y n in positive integers x, y, and n, with n ≥ 3, and 5 ≤ m ≤ 50. We reduce the problem to a study of systems of binomial Thue equations, and use a combination of local arguments, the modular method via Frey curves, and bounds arising from linear forms in logarithms to solve the problem.

D(n)-quintuples with square elements

arXiv (Cornell University), Nov 3, 2020

For an integer n, a set of m distinct nonzero integers {a 1 , a 2 , . . . , am} such that a i a j... more For an 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, we show that there are infinitely many essentially different D(n)-quintuples with square elements. We obtained this result by constructing genus one curves on a certain double cover of A 2 branched along four curves.

On fundamental units of real quadratic fields of class number 1

Archiv der Mathematik, Jun 29, 2019

In this paper, we give a nontrivial lower bound for the fundamental unit of norm -1 of a real qua... more

On the size of Diophantine m-tuples

arXiv (Cornell University), Feb 11, 2000

Let n be a nonzero integer and assume that a set S of positive integers has the property that xy ... more 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.

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, ... 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.

Number Theory, Školska knjiga, Zagreb, 2021

Number theory is a branch of mathematics that is primarily focused on the study of positive integ... 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.