Andrej Dujella:

Diophantine m-tuples and elliptic curves

Graduate course     (2021/2022)

Course description

A rational Diophantine m-tuple a is set of m non-zero rational numbers with the property that the product of any two of them increased by 1 is a perfect square. The problem of extension of a rational Diophantine triple {a,b,c} to a quadruple leads naturally to studding the elliptic curve y2 = (ax+1)(bx+1)(cx+1). This connection between Diophantine m-tuples and elliptic curves prove to be very fruitful. In particular, it is used in the recent construction of infinitely many rational Diophantine sextuples, which has been an open problem since the time when Euler proved that there are infinitely many rational Diophantine quintuples (his construction also has an elegant interpretation in terms of elliptic curves). On the other hand, curves obtained using Diophantine triples and quadruples were used in the construction of elliptic curves with a given torsion group and large rank, so even today some of the record ranks for elliptic curves over Q and Q(t) are obtained by this method. The course will deal with the mentioned connections of elliptic curves with rational Diophantine m-tuples, and also with integer Diophantine m-tuples, where questions of possible torsion groups and integer points on such curves arise.

Topics to be covered in this course include: a review of the basic results on integer and rational Diophantine m-tuples; introduction to elliptic curves over the field of rational numbers; Mordell-Weil group of elliptic curves induced by rational Diophantine triples; construction of infinite families of rational Diophantine quintuples and sextuples; general methods for calculating rank and construction of elliptic curves of large rank; application of rational Diophantine triples in the construction of high-rank curves over Q and Q(t) with torsion groups Z/2Z × Z/2Z, Z/2Z × Z/4Z, Z/2Z × Z/6Z and Z/2Z × Z/8Z; elliptic curves with rank 0; possible torsion groups of elliptic curves induced by integer Diophantine triples; integer points on some families of elliptic curves; D(q)-m-tuples and elliptic curves; strong Diophantine triples; elliptic curves induced by rational Diophantine quadruples. Some additional related topics will be covered through student seminar lectures.

It will be assumed that the students are familiar with the basic notions and results from number theory, at the level covered in the undergraduate course Number Theory.


  1. A. Dujella: Teorija brojeva, Skolska knjiga, Zagreb, 2019.

  2. A. Dujella: Number Theory, Skolska knjiga, Zagreb, 2021.

  3. A. Filipin, Z. Franusic: Diofantovi skupovi, University of Zagreb, lecture notes, 2020.

  4. J. H. Silverman, J. Tate: Rational Points on Elliptic Curves, Springer-Verlag, Berlin, 1992.

  5. L. C. Washington: Elliptic Curves: Number Theory and Cryptography, CRC Press, Boca Raton, 2008.

  6. Selected recent papers from

Lecture notes
(in pdf format; in Croatian)

Seminar topics

Homework exercises:

ex1   ex2   ex3   ex4   ex5  

Some (useful) links

Seminar on Number Theory and Algebra (University of Zagreb)
Number Theory - Undergraduate course
Elliptic Curves in Cryptography - Graduate course
Diophantine m-tuples page
High rank elliptic curves with prescribed torsion
Infinite families of elliptic curves with high rank and prescribed torsion
History of elliptic curves rank records
High rank elliptic curves with prescribed torsion over quadratic fields
PARI/GP home page
PARI/GP in browser
MAGMA Calculator
Elliptic curves in GeoGebra (Sime Suljic)
Number Theory Web
Number theory groups and seminars
Recommended readings for graduate students in number theory

Andrej Dujella home page