# Mirela Jukić Bokun

Assistant Professor
Department of Mathematics
Josip Juraj Strossmayer University of Osijek
Trg Ljudevita Gaja 6
Osijek, HR-31000, Croatia¸
 phone: +385-31-224-842 fax: +385-31-224-801 email: mirela @ mathos.hr office: 15 (ground floor)

## Research Interests

Number Theory (Elliptic Curves, Diophantine Equations)

## Degrees

PhD in Mathematics, Department of Mathematics, University of Zagreb, 2011.
MSc in Mathematics, Department of Mathematics, University of Zagreb, Croatia, 2008.
BSc in Mathematics and Computer Science, Department of Mathematics, University of Osijek, Croatia, 2001.

## Publications

Journal Publications

1. M. Jukić Bokun, I. Soldo, Pellian equations of special type, Mathematica Slovaca (2021), prihvaćen za objavljivanje
In this paper, we consider the solvability of the Pellian equation x^2-(d^2+1)y^2=-m, in cases d=n^k, m=n^{2l-1}, where k,l are positive integers, n is a composite positive integer and d=pq, m=pq^2, p,q are primes. We use the obtained results to prove results on the extendibility of some D(-1)-pairs to quadruples in the ring Z[sqrt{-t}], with t>0.
2. A. Filipin, M. Jukić Bokun, I. Soldo, On $D(-1)$-triples ${1,4p^2+1,1-p}$ in the ring $Z[sqrt{-p}]$ with a prime $p$, Periodica Mathematica Hungarica (2020), prihvaćen za objavljivanje
Let $p$ be a prime such that $4p^2+1$ is also a prime. In this paper, we prove that the $D(-1)$-set ${1,4p^2+1,1-p}$ cannot be extended with the forth element $d$ such that the product of any two distinct elements of the new set decreased by $1$ is a square in the ring $Z[sqrt{-p}]$.
3. A. Dujella, M. Jukić Bokun, I. Soldo, A Pellian equation with primes and applications to D(−1)-quadruples, Bulletin of the Malaysian Mathematical Sciences Society 42 (2019), 2915-2926
In this paper, we prove that the equation x^2 − (p^(2k+2) + 1)y^2 = −p^(2l+1), l∈{0, 1, . . . , k}, k ≥ 0, where p is an odd prime number, is not solvable in positive integers x and y. By combining that result with other known results on the existence of Diophantine quadruples, we are able to prove results on the extensibility of some D(−1)-pairs to quadruples in the ring Z[√−t], t > 0.
4. M. Jukić Bokun, I. Soldo, On the extensibility of D(-1)-pairs containing Fermat primes, Acta Mathematica Hungarica 159 (2019), 89-108
In this paper, we study the extendibility of a D(-1)-pair {1,p}, where p is a Fermat prime, to a D(-1)-quadruple in Z[sqrt{-t}], t>0.
5. A. Dujella, M. Jukić Bokun, I. Soldo, On the torsion group of elliptic curves induced by Diophantine triples over quadratic fields, RACSAM 111 (2017), 1177-1185
The possible torsion groups of elliptic curves induced by Diophan- tine triples over quadratic fields, which do not appear over Q, are Z/2Z × Z/10Z, Z/2Z × Z/12Z and Z/4Z × Z/4Z. In this paper, we show that all these torsion groups indeed appear over some quadratic field. Moreover, we prove that there are infinitely many Diophantine triples over quadratic fields which induce elliptic curves with these tor- sion groups.
6. J. Aguirre, A. Dujella, M. Jukić Bokun, J.C. Peral, High rank elliptic curves with prescribed torsion group over quadratic fields, Periodica Mathematica Hungarica 68 (2014), 222-230
There are 26 possibilities for the torsion group of elliptic curves defined over quadratic number fields. We present examples of high rank elliptic curves with given torsion group which give the current records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for Z/15Z, there exist an elliptic curve over some quadratic field with this torsion group and with rank >= 2.
7. M. Jukić Bokun, Elliptic curves over quadratic fields with fixed torsion subgroup and positive rank, Glasnik Matematički 47 (2012), 277-284
For each of the torsion groups Z/2ZxZ/10Z, Z/2ZxZ/12Z, Z/15Z we find the quadratic field with the smallest absolute value of its discriminant such that there exists an elliptic curve with that torsion and positive rank. For the torsion groups Z/11Z, Z/14Z we solve the analogous problem after assuming the Parity conjecture.
8. M. Jukić Bokun, On the rank of elliptic curves over Q(sqrt{-3}) with torsion groups Z/3Z x Z/3Z and Z/3Z x Z/6Z, Proceedings of the Japan Academy. Series A Mathematical sciences 87/5 (2011), 61-64
We construct elliptic curves over the field Q(sqrt{-3}) with torsion group Z/3Z x Z/3Z and ranks equal to 7 and an elliptic curve over the same field with torsion group Z/3Z x Z/6Z and rank equal to 6.
9. A. Dujella, M. Jukić Bokun, On the rank of elliptic curves over Q(i) with torsion group Z/4Z × Z/4Z, Proceedings of the Japan Academy. Series A Mathematical sciences 86/6 (2010), 93-96
We construct an elliptic curve over Q(i) with torsion group Z/4Z * Z/4Z and rank equal to 7 and a family of elliptic curves with the same torsion group and rank >= 2.

Others

1. M. Jukić Bokun, A. Behin, Eulerova funkcija, Math.e : hrvatski matematički elektronski časopis 31 (2017)
2. M. Đumić, M. Jukić Bokun, Euklidov algoritam, Osječki matematički list 13 (2013), 121-137
3. M. Duk, M. Jukić Bokun, L'Hospitalovo pravilo, Poučak 51 (2012), 19-31
4. M. Jukić, Apolonijev problem, Osječki matematički list 2/2 (2002), 81-90
5. M. Jukić, Broj e, Osječki matematički list 1/2 (2001), 79-85

## Projects

• GAMe-based learning in MAthematics (GAMMA)

Coordinator of the project. Erasmus+ project (Key action: KA2 - Cooperation for innovation and the exchange of good practices, Action Type: KA201: Strategic Partnership for school education).

• Diophantine m-tuples, elliptic curves, Thue and index form equations

Project leader: prof. dr. Andrej Dujella, Department of Mathematics, University of Zagreb. Project by the Croatian Science Foundation for period 2014.-2018.

• Passive control of mechanical models

Project leader: prof. dr. Ninoslav Truhar, Department of Mathematics, University of Osijek. Scientific project No.235-2352818-1042 of the Croatian Ministry of Science, Education and Sports for the period 2007.-2013. (junior researcher)

• Statistical aspects of parameter identification problems

Project leader: prof. dr. Mirta Benšić, Department of Mathematics, University of Osijek. Scientific project No. 0235002 of the Croatian Ministry of Science, Education and Sports for the period 2002.-2006. (junior researcher)

## Professional Activities

Editorial Boards
• Since 2012. technical editor of the international journal Mathematical Communications

Committee Memberships

Refereeing/Reviewing

Refereeing:

• Osječki matematički list

Workshops and Conferences

with talk:

• Elliptic curves over quadratic fields with fixed torsion subgroup and positive rank, 5th Croatian Mathematical Congress, Rijeka, Croatia, June 18 - 21, 2012
• High rank elliptic curves with prescribed torsion group over quadratic fields, Workshop on Number Theory and Algebra, Department of Mathematics, University of Zagreb, Zagreb, Croatia, November 26-28, 2014
• On the torsion group of elliptic curves induced by Diophantine triples over quadratic fields, 1st Croatian Combinatorial Days, Zagreb, Croatia, September 29-30, 2016
• On the torsion group of elliptic curves induced by Diophantine triples over quadratic fields, XXXth Journées Arithmétiques, Caen, France, July 3-7, 2017
• A Pellian equation with primes and its applications, Fibonacci Conference, Sarajevo, Bosna i Hercegovina, July 21-23 2020.
• On the extensibility of some parametric families of D(-1)-pairs to quadruples in the ring Z[\sqrt{;-t};], t>0, 9th International Eurasian Conference on Mathematical Sciences and Applications, Skopje, North Macedonia, August 25-28, 2020.

without talk:

• Winter School on Explicit Methods in Number Theory, Debrecen, Hungary, January 26-30, 2009

Seminar Talks
• Mestre's polynomial method for constructing elliptic curves of large rank, March 29, 2006, Seminar on Number Theory and Algebra, Department of Mathematics, University of Zagreb
• Some methods for constructing elliptic curves of large rank, April 2, 2008, Seminar on Number Theory and Algebra, Department of Mathematics, University of Zagreb
• Elliptic curves with nontrivilal torsion group and large rank, April 16, 2008, Seminar on Number Theory and Algebra, Department of Mathematics, University of Zagreb
• Elliptic curves of large rank over quadratic fields I, February 2, 2011,  Seminar on Number Theory and Algebra, Department of Mathematics, University of Zagreb
• Elliptic curves of large rank over quadratic fields II, May 25, 2011,  , Department of Mathematics, University of Zagreb

Service Activities

## Teaching

Konzultacije (Office Hours): Po dogovoru.

Prijedlog tema za diplomske i završne radove

Financijska i aktuarska matematika

Primijenjena matematika za računalnu znanost

Teorija brojeva

Nastavne aktivnosti u prošlosti (Past Courses)

Metodika nastave matematike II, Analitička geometrija, Algebra Integralni račun Obične diferencijalne jednadžbe, Geometrija prostora i ravnine, Elementarna matematika I, Elementarna matematika II, Linearna algebra I, Linearna algebra II, Linearna algebra III  (Odjel za matematiku)

Matematika (Ekonomski fakultet)

Matematika III, Matematička analiza II, Linearna algebra (Elektrotehnički fakultet)

Matematika (Poljoprivredni fakultet)

21-23 July 2020