Ivan Soldo
Associate Professor Department of Mathematics Josip Juraj Strossmayer University of Osijek Trg Ljudevita Gaja 6, HR31 000 Osijek, Croatia

Research Interests
Number Theory, i.e. diophantine equations over imaginary quadratic fields and diophantine mtuples.
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
 Y. Fujita, I. Soldo, D(−1)tuples in the ring Z[√−k] with k > 0, Publicationes Mathematicae 100 (2022), 4967Let n be a nonzero integer and R a commutative ring. A D(n)mtuple in R is a set of m nonzero elements in R such that the product of any two distinct elements plus n is a perfect square in R. In this paper, we prove that there does not exist a D(−1)quadruple {a, b, c, d} in the ring Z[√−k], k ≥ 2 with positive integers a
 A. Filipin, M. Jukić Bokun, I. Soldo, On $D(1)$triples ${1,4p^2+1,1p}$ in the ring $Z[sqrt{p}]$ with a prime $p$, Periodica Mathematica Hungarica 85 (2022), 292302Let $p$ be a prime such that $4p^2+1$ is also a prime. In this paper, we prove that the $D(1)$set ${1,4p^2+1,1p}$ 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 $Z[sqrt{p}]$.
 M. Jukić Bokun, I. Soldo, Pellian equations of special type, Mathematica Slovaca 71/6 (2021), 15991607In 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^{2l1}, 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.
 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), 29152926In 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.
 M. Jukić Bokun, I. Soldo, On the extensibility of D(1)pairs containing Fermat primes, Acta Mathematica Hungarica 159 (2019), 89108In 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.
 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), 645657According 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 wellknown 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.
 A. Dujella, M. Jukić Bokun, I. Soldo, On the torsion group of elliptic curves induced by Diophantine triples over quadratic fields, RACSAM 111 (2017), 11771185The 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.
 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), 12011224In 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].
 Z. Franušić, I. Soldo, The problem of Diophantus for integers of Q[√ 3], Rad HAZU, Matematičke znanosti. 18 (2014), 1525We 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.
 I. Soldo, On the existence of Diophantine quadruples in Z[√ 2], Miskolc Mathematical Notes 14/1 (2013), 265277By the work of Abu Muriefah, AlRashed, 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.
 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), 296330Let 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].
 A. Dujella, I. Soldo, Diophantine quadruples in Z[sqrt(2)], Analele Stiintifice ale Universitatii Ovidius Constanta Seria Matematica 18/3 (2010), 8198In 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, AlRashed 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.
 I. Soldo, K. Vincetić, Cjelobrojne funkcijske jednadžbe, Matematičko fizički list 67/2 (2016), 93103Funkcijske jednadžbe su jednadžbe u kojima je nepoznanica funkcija. Rješenje takve jednadžbe je svaka funkcija koja ju zadovoljava. U radu ćemo prikazati i primjerima potkrijepiti neke metode za rješavanje funkcijskih jednadžbi s jednom i dvije nezavisne varijable.
 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 DANE : ShapleyShubik indeks, Banzhaf indeks i DeeganPackel indeks. Za ilustraciju tih indeksa navedeno je nekoliko primjera. Web strana: www.glasslavonije.hr/sglasnik/sveucilisniglasnik18.pdf
 I. Soldo, I. Vuksanović, Pitagorine trojke, Matematičko fizički list 255 (2014), 179184
 I. Soldo, I. Mandić, Pellova jednadžba, Osječki matematički list 8 (2008), 2936Članak 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.
 I. Soldo, Različiti načini množenja matrica, Osječki matematički list 5 (2005), 18U članku se analiziraju različiti načini množenja matrica. Svaki od njih ilustriran je primjerom.
 K. Burazin, J. Jankov, I. Kuzmanović Ivičić, 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, D(1)tuples in the ring Z[√−k] with k>0, Conference on Diophantine mtuples and related problems III, September 14 – 16, 2022, Faculty of Civil Engineering, University of Zagreb, Zagreb, Croatia
 I. Soldo, D(1)tuples in the ring Z[√−k] with k>0, 7th Croatian Mathematical Congress, June 1518, 2022, Faculty of Science, University of Split, Croatia
 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 68, 2019, Bursa, Turkey.
 I. Soldo, A Pellian equation in primes and its applications, Representation Theory XVI, June 2429, 2019, Dubrovnik, Croatia.
 I. Soldo, Applications of a Diophantine equation of a special type, Conference on Diophantine mtuples and Related Problems II, October 1517, 2018, Purdue University Northwest, Westville/Hammond, Indiana, USA
 I. Soldo, A Pellian equation with primes and its applications, XXX^{th} Journées Arithmétiques, July 37, 2017, Caen, France.
 T. Marošević, I. Soldo, Indices of political power–a case study of a few parliaments, 16^{th} International Conference on Operational Research KOI 2016, September 2729, 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 1519, 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 2628, 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 mtuples and related problems, November 1315, 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 15, 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

20142018 member of the scientific project entitled with Diophantine mtuples, elliptic curves Thue and index of equations (supported by Croatian Science Foundation).
 20182022 member of the scientific project entitled with Diophantine geometry and applications (supported by Croatian Science Foundation).
Postgraduate Seminars
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 BoardsSince 2009, technical editor of the international Journal Mathematical Communications.
Refereeing/Reviewing
One of the referees of a Journal Osječki matematički list.
Commite Memberships
 Member of the Organize Committee of the 4^{th} Croatian Congress of Mathematics, Osijek, 2008
 Member of the Organize Committee of the 15^{th} International Conference on Operational Research, Croatian Operational Research Society, Osijek 2014
 Member of the Organize Committee of the Workshop on Number Theory and Algebra, Zagreb, 2014
Service Activities
Since 2012, secretary of the Craotian Operationl Researsh Society
Teaching
Diferencijalni račun (zimski semestar)
 utorak (Tue), 8:00  12:00, P 1
Kriptografija (zimski semestar)
 srijeda (Wed), 14:00  18:00, P 2
Integralni račun (ljetni semestar)
Konzultacije (Office Hours): Utorak (Thu) 11:00pm12:00pm; konzultacije su moguće i nakon održane nastave, a i po dogovoru.