# Ivan Soldo

Assistant Professor
Department of Mathematics
Josip Juraj Strossmayer University of Osijek
Trg Ljudevita Gaja 6,
HR-31 000 Osijek, Croatia
 phone: +385-31-224-841 fax: +385-31-224-801 email: Ova e-mail adresa je zaštićena od spambota. Potrebno je omogućiti JavaScript da je vidite. office: 14 (ground floor)

## Research Interests

Number Theory, i.e. diophantine equations over imaginary quadratic fields and diophantine m-tuples.

## Degrees

B.Sc. February 17, 2005, Department of Mathematics, University of Osijek, Croatia.
PhD, July 2, 2012, Department of Mathematics, University of Zagreb, Croatia

## Publications

Journal Publications

1. 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}]\$.
2. 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.
3. 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.
4. T. Marošević, I. Soldo, Modified indices of political power: a case study of a few parliaments, Central European Journal of Operations Research 26/3 (2018), 645-657
According to yes–no voting systems, players (e.g., parties in a parliament) have some inﬂuence on making some decisions. In formal voting situations, taking into account that a majority vote is needed for making a decision, the question of political power of parties can be considered. There are some well-known indices of political power e.g., the Shapley–Shubik index, the Banzhaf index, the Johnston index, the Deegan–Packel index. In order to take into account different political nature of the parties, as the main factor for forming a winning coalition i.e., a parliamentary majority, we give a modiﬁcation of the power indices. For the purpose of comparison of these indices of political power from the empirical point of view, we consider the indices of power in some cases, i.e., in relation to a few parliaments.
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. I. Soldo, D(-1)-triples of the form {1, b, c} in the ring Z[√ -t], t>0, Bulletin of the Malaysian Mathematical Sciences Society 39/3 (2016), 1201-1224
In this paper, we study D(-1)-triples of the form {1, b, c} in the ring Z[√ -t], t>0, for positive integer b such that b is a prime, twice prime and twice prime squared. We prove that in those cases c has to be an integer. In cases of b=26, 37 or 50 we prove that D(-1)-triples of the form {1, b, c} cannot be extended to a D(-1)-quadruple in the ring Z[√ -t], t>0, except in cases t in {1, 4, 9, 25, 36, 49}. For those exceptional cases of t we show that there exist infinitely many D(-1)-quadruples of the form {1, b, -c, d}, c, d>0 in Z[√ -t].
7. Z. Franušić, I. Soldo, The problem of Diophantus for integers of Q[√ -3], Rad HAZU, Matematičke znanosti. 18 (2014), 15-25
We solve the problem of Diophantus for integers of the quadratic field Q[√ -3] by finding a D(z)-quadruple in Z[(1+√ -3)/2] for each z that can be represented as a difference of two squares of integers in Q[√ -3], up to finitely many possible exceptions.
8. I. Soldo, On the existence of Diophantine quadruples in Z[√ -2], Miskolc Mathematical Notes 14/1 (2013), 265-277
By the work of Abu Muriefah, Al-Rashed, Dujella and the author, the problem of the existence of D(z)-quadruples in the ring Z[√ -2] has been solved, except for the cases z=24a+2+(12b+6)√ -2, z=24a+5+(12b+6)√ -2, z=48a+44+(24b+12)√ -2. In this paper, we present some new formulas for D(z)-quadruples in these remaining cases, involving some congruence conditions modulo 11 on integers a and b. We show the existence of D(z)-quadruple for significant proportion of the remaining three cases.
9. I. Soldo, On the extensibility of D(-1)-triples {1, b, c} in the ring Z[√ -t], t > 0, Studia scientiarum mathematicarum Hungarica 50/3 (2013), 296-330
Let b = 2, 5, 10 or 17 and t > 0. We study the existence of D(-1)-quadruples of the form {1, b, c, d} in the ring Z[√ -t]. We prove that if {1, b, c} is a D(-1)-triple in Z[√ -t], then c is an integer. As a consequence of this result, we show that for t otin {1, 4, 9, 16} there does not exist a subset of Z[√ -t] of the form {1, b, c, d} with the property that the product of any two of its distinct elements diminished by 1 is a square of an element in Z[√ -t].
10. A. Dujella, I. Soldo, Diophantine quadruples in Z[sqrt(-2)], Analele Stiintifice ale Universitatii Ovidius Constanta Seria Matematica 18/3 (2010), 81-98
In this paper, we study the existence of Diophantine quadruples with the property D(z) in the ring Z[sqrt(-2)]. We find several new polynomial formulas for Diophantine quadruples with the property D(a+b*sqrt(-2)), for integers a and b satisfying certain congruence conditions. These formulas, together with previous results on this subject by Abu Muriefah, Al-Rashed and Franusic, allow us to almost completely char- acterize elements z of Z[sqrt(-2)] for which a Diophantine quadruple with the property D(z) exists.

Others

1. I. Soldo, K. Vincetić, Cjelobrojne funkcijske jednadžbe, Matematičko fizički list 67/2 (2016), 93-103
Funkcijske jednadžbe su jednadžbe u kojima je nepoznanica funkcija. Rješenje takve jednadžbe je svaka funkcija koja ju zadovoljava. U radu ćemo prikazati i primjerima potkrijepiti neke metode za rješavanje funkcijskih jednadžbi s jednom i dvije nezavisne varijable.
2. T. Marošević, I. Soldo, Kako se mjeri snaga stranaka u parlamentu (2016)
U članku su prikazani neki kvantitativni (brojčani) pokazatelji političke snage u sustavu glasovanja DA-NE : Shapley-Shubik indeks, Banzhaf indeks i Deegan-Packel indeks. Za ilustraciju tih indeksa navedeno je nekoliko primjera. Web strana: www.glas-slavonije.hr/sglasnik/sveucilisni-glasnik-18.pdf
3. I. Soldo, I. Vuksanović, Pitagorine trojke, Matematičko fizički list 255 (2014), 179-184
4. I. Soldo, I. Mandić, Pellova jednadžba, Osječki matematički list 8 (2008), 29-36
Članak sadrži riješene primjere i probleme koji se svode na analizu skupa rješenja Pellove jednadžbe x^2 - dy^2 = 1 te njenu usku povezanost sa diofantskim aproksimacijama i veržnim razlomcima.
5. I. Soldo, Različiti načini množenja matrica, Osječki matematički list 5 (2005), 1-8
U članku se analiziraju različiti načini množenja matrica. Svaki od njih ilustriran je primjerom.

Books

1. K. Burazin, J. Jankov, I. Kuzmanović, I. Soldo, Primjene diferencijalnog i integralnog računa funkcija jedne varijable, Sveučilište Josipa Jurja Strossmayera u Osijeku - Odjel za matematiku, Osijek, 2017.

## Conference Talks and Participations

• I. Soldo, On the extensibility of some parametric families of D(-1)-pairs to quadruples in the rings of integers of the imaginary quadratic fields, Friendly workshop on diophantine equations and related problems, July 6-8, 2019, Bursa, Turkey.
• I. Soldo, A Pellian equation in primes and its applications, Representation Theory XVI, June 24-29, 2019, Dubrovnik, Croatia.
• I. Soldo, Applications of a Diophantine equation of a special type, Conference on Diophantine m-tuples and Related Problems II, October 15-17, 2018, Purdue University Northwest, Westville/Hammond, Indiana, USA
• I. Soldo, A Pellian equation with primes and its applications, XXXth Journées Arithmétiques, July 3-7, 2017,  Caen, France.
• T. Marošević, I. Soldo, Indices of political power–a case study of a few parliaments, 16th International Conference on Operational Research KOI 2016, September 27-29, 2016, Osijek, Croatia.
• I. Soldo, Diophantine triples in the ring of integers of the quadratic field Q(√-t), t>0, Computational Aspects of  Diophantine equations, February 15-19, 2016, Salzburg, Austria.
• I. Soldo, D(-1)-triples of the form {1,b,c} in the ring Z[√-t], t>0, Workshop on Number Theory and Algebra, Department of Mathematics, University of Zagreb, November 26-28, 2014, Zagreb, Croatia.
• I. Soldo, D(-1)-triples of the form {1,b,c} and their extensibility in the ring Z[√-t], t>0, Conference on Diophantine m-tuples and related problems, November 13-15, 2014, Purdue University North Central, Westville, Indiana, USA.
• I. Soldo, D(z)-quadruples in the ring Z[√-2], for some exceptional cases o z, Erdös Centennial, July 1-5, 2013, Budapest, Hungary.
• I. Soldo, The problem of existence of Diophantine quadruples in Z[√-2],  5th Croatian Mathematical Congress, June 18 - 21, 2012, Rijeka, Croatia.
• I. Soldo, Diophantine quadruples in Z[√-2], Number Theory and Its Applications, An International Conference Dedicated to Kálman Győry, Attila Pethő, János Pintz and András Sárközy, Debrecen, Hungary, 2010.
• Winter School on Explicit Methods in Number Theory, January 26 - 30, 2009, Debrecen, Hungary
• 4th Croatian Mathematical Congress, June 17 - 20, 2008, Osijek, Croatia.
• Conference from Diophantine Approximations, July 25 - 27, 2007, Graz, Austria.
• K. Sabo, I. Soldo, Računanje udaljenosti točke do krivulje, Zbornik radova PrimMath[2003],
Mathematica u znanosti, tehnologiji i obrazovanju, September 25 - 26, 2003., pp. 215 - 225.

## Projects

• 2014-2018 member of the scientific project entitled with Diophantine m-tuples, elliptic curves Thue and index of equations (supported by Croatian Science Foundation).

• 2018-2022 member of the scientific project entitled with Diophantine geometry and applications (supported by Croatian Science Foundation).

Member of Seminar for Number Theory and Algebra
Leaders are prof.dr.sc. A. Dujella and prof.dr.sc. I. Gusić.

• The extensibility of some D(-1)-triples over the imaginary quadratic fields II, April 25, 2012.
• The extensibility of some D(-1)-triples over the imaginary quadratic fields I, April 11, 2012.
• Diophantine quadruples in Z[√-2], September 22, 2010.
• General Weierstrass equation, June 18, 2008.
• Criteria for solvability of Pellian equation x^2- d y^2 = +-2, June 20, 2007.
• Numerical computation of zeroes of Riemann Zeta function, April 20, 2007.

## Professional Activities

Editorial Boards

Since 2009, technical editor of the international Journal Mathematical Communications.

Refereeing/Reviewing

One of the referees of a Journal Osječki matematički list.

Commite Memberships

## Service Activities

Since 2012, secretary of the Craotian Operationl Researsh Society

## Teaching

Diferencijalni račun (zimski semestar)

utorak (Tue), 8:00 - 12:00, P 2

Kriptografija i sigurnost sustava (zimski semestar)

petak (Fri), 8:00 - 12:00, P 3

MatematikaFakultet agrobiotehničkih znanosti(zimski semestar)

Integralni račun (ljetni semestar)

Uvod u teoriju brojeva (ljetni semestar)

Matematika II, Odjel za fiziku (ljetni semestar)

Konzultacije (Office Hours): Utorak (Thu) 11:00pm-12:00pm;  konzultacije su moguće i nakon održane nastave, a i po dogovoru.