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

1. M. Jukić Bokun, I. Soldo, Extensions of D(-1)-pairs in some imaginary quadratic fields, New York Journal of Mathematics 30 (2024), 745-755
   Abstract

   In this paper, we discuss the extensibility of Diophantine D(-1) pairs {a, b}, where a,b are positive integers in the ring Z[\sqrt{-k}], k>0. We prove that families of such D(-1)-pairs with b=p^i q^j, where p,q are different odd primes and i,j are positive integers cannot be extended to quadruples in certain rings Z[\sqrt{-k}], where k depends on p^i, q^i and a. Further, we present the result on non-existence of D(-1)-quintuples of a specific form in certain imaginary quadratic rings.
2. M. Jukić Bokun, I. Soldo, Triangular D(−1)-tuples, Bulletin Mathematique de la Société des Sciences Mathématiques de Roumanie (2024), prihvaćen za objavljivanje
   Abstract

   In this paper, we consider the extendibility of the triangular D(−1)-tuples, i.e., the sets of the positive integers with the property that the product of any two of them decreased by 1 is the triangular number. We prove that the only triangular D(−1)-triples of the form {1, 2, c}, c = 2^np, where n is a non-negative integer and p is a prime, are those with c ∈ {11, 46, 352, 11936}. In addition, we prove that for these c’s no triangular D(−1)-quadruple of the form {1, 2, c, d} exists.
3. Y. Fujita, I. Soldo, On the extendibility of certain $D(-1)$-pairs in imaginary quadratic rings, Indian Journal of Pure and Applied Mathematics 1 (2023)
   Abstract

   Let $R$ be a commutative ring with unity $1$. A set of $m$ different non-zero elements in $R$ such that the product of any two distinct elements decreased by $1$ is a perfect square in $R$ is called a $D(-1)$-$m$-tuple in $R$. In the ring $bZ[sqrt{-k}]$, with an integer $kge 2$, we consider the $D(-1)$-pairs ${a,2^i p^j}$, where $iin{0,1}$, $a, j$ are positive integers, $p$ is an odd prime, $gcd(a, 2^i p^j)=1$ and $a<2^i p^j$. We prove that there does not exist a $D(-1)$-quadruple of the form ${a, 2^i p^j,c,d}$ in $bZ[sqrt{-k}]$ in the following cases: $k$ does not divide $2^i p^j-a$; $k$ divides $2^i p^j-a$ and $(2^i p^j-a)/k$ is a prime; $k=2^i p^j-a$ and $a>1$.
4. Y. Fujita, I. Soldo, The non-existence of $D(-1)$-quadruples extending certain pairs in imaginary quadratic rings, Acta Mathematica Hungarica 170/2 (2023), 455-482
   Abstract

   A $D(n)$-$m$-tuple, where $n$ is a non-zero integer, is a set of $m$ distinct elements in a commutative ring $R$ 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 $bZ[sqrt{-k}]$, $kge 2$ with positive integers $a<b< 16a^2-a-2+2sqrt{k(8a^2+3a+1)}$ and integers $c$ and $d$ satisfying $d<0<c$. By combining that result with [14, Theorem 1.1], we were able to obtain a general result on the non-existence of a $D(-1)$-quadruple ${a,b,c,d}$ in $bZ[sqrt{-k}]$ with integers $a,b,c,d$ satisfying $a<ble 8a-3$. Furthermore, for a non-negative integer $i$ and a positive integer $j$, we apply the obtained results in proving of the non-existence of $D(-1)$-quadruples containing powers of primes $p^i$, $q^j$ with an arbitrary different primes $p$ and $q$.
5. Y. Fujita, I. Soldo, D(−1)-tuples in the ring Z[√−k] with k > 0, Publicationes Mathematicae 100 (2022), 49-67
   Abstract

   Let n be a non-zero integer and R a commutative ring. A D(n)-m-tuple in R is a set of m non-zero 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 <b ≤ 8a−3 and negative integers c and d. By using that result we were able to prove that such a D(−1)-pair {a, b} cannot be extended to a D(−1)- quintuple {a, b, c, d, e} in Z[√−k] with integers c, d and e. Moreover, we apply the obtained result to the D(−1)-pair {p^i, q^j} with an arbitrary different primes p, q and positive integers i, j.

6. 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 85 (2022), 292-302
   Abstract

   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 $Z[sqrt{-p}]$.
7. M. Jukić Bokun, I. Soldo, Pellian equations of special type, Mathematica Slovaca 71/6 (2021), 1599-1607
   Abstract

   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.
8. 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
   Abstract

   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.
9. M. Jukić Bokun, I. Soldo, On the extensibility of D(-1)-pairs containing Fermat primes, Acta Mathematica Hungarica 159 (2019), 89-108
   Abstract

   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.
10. 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
    Abstract

    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.
11. 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
    Abstract

    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.
12. 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
    Abstract

    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].
13. Z. Franušić, I. Soldo, The problem of Diophantus for integers of Q[√ -3], Rad HAZU, Matematičke znanosti. 18 (2014), 15-25
    Abstract

    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.
14. I. Soldo, On the existence of Diophantine quadruples in Z[√ -2], Miskolc Mathematical Notes 14/1 (2013), 265-277
    Abstract

    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.
15. 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
    Abstract

    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].
16. A. Dujella, I. Soldo, Diophantine quadruples in Z[sqrt(-2)], Analele Stiintifice ale Universitatii Ovidius Constanta Seria Matematica 18/3 (2010), 81-98
    Abstract

    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.

1. M. Andrijević, I. Soldo, Neke metode faktorizacije prirodnih brojeva, Osječki matematički list 23/1 (2023), 1-11
   Abstract

   Faktorizacija prirodnih brojeva u praksi može biti vrlo zahtjevna. Jedna od najčešćih primjena je u dešifriranju kriptosustava s javnim ključem, kao što je primjerice RSA kriptosustav. U ovome članku prezentiramo neke od neuobičajenih metoda faktorizacije kao što su Fermatova metoda i metoda verižnog razlomka za faktorizaciju velikih prirodnih brojeva.
2. I. Soldo, K. Vincetić, Cjelobrojne funkcijske jednadžbe, Matematičko fizički list 67/2 (2016), 93-103
   Abstract

   Funkcijske 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.
3. T. Marošević, I. Soldo, Kako se mjeri snaga stranaka u parlamentu (2016)
   Abstract

   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
4. I. Soldo, I. Vuksanović, Pitagorine trojke, Matematičko fizički list 255 (2014), 179-184
5. I. Soldo, I. Mandić, Pellova jednadžba, Osječki matematički list 8 (2008), 29-36
   Abstract

   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.
6. I. Soldo, Različiti načini množenja matrica, Osječki matematički list 5 (2005), 1-8
   Abstract

   U članku se analiziraju različiti načini množenja matrica. Svaki od njih ilustriran je primjerom.

1. M. Jukić Bokun, I. Soldo, Zbirka zadataka iz teorije brojeva, Fakultet primijenjene matematike i informatike, Osijek, 2023.
2. K. Burazin, J. Jankov Pavlović, I. Kuzmanović Ivičić, I. Soldo, Primjene diferencijalnog i integralnog računa funkcija jedne varijable, Sveučilište Josipa Jurja Strossmayera u Osijeku - Odjel za matematiku, Osijek, 2017.

Projects

• 2023-2027 member of the scientific project entitled with Number Theory and Arithmetic Geometry (supported by Croatian Science Foundation)
• 2018-2022 member of the scientific project entitled with Diophantine geometry and applications (supported by Croatian Science Foundation)
• 2014-2018 member of the scientific project entitled with Diophantine m-tuples, elliptic curves Thue and index of equations (supported by Croatian Science Foundation).

Professional Activities

Conference talks and participations 

• I. Soldo, Diophantine m-tuples in certain imaginary quadratic fields, 8th Croatian Mathematical Congress, July 2-5, 2024, School of Applied Mathematics and Informatics, Osijek, Croatia
• I. Soldo, D(−1)-quadruples extending certain pairs in imaginary quadratic rings, Modular curves and Galois representations, September 18 – 22, 2023, Department of Mathematics, Faculty of Science, University of Zagreb, Zagreb, Croatia
• I. Soldo, D(-1)-tuples in the ring Z[√−k] with k>0, Conference on Diophantine m-tuples 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 15-18, 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 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, XXX^th Journées Arithmétiques, July 3-7, 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 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.

Editorial Boards

Technical editor of the international Journal Mathematical Communications (since 2009).

Commite Memberships

Member of Seminar for Number Theory and Algebra

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 16^th International Conference on Operational Research, Croatian Operational Research Society, Osijek 2016

Member of the Organize Committee of the Workshop on Number Theory and Algebra, Zagreb, 2014

Member of the Organize Committee Conference on Diophantine m-tuples and related problems III, Zagreb 2022. 

Member of the Organize Committee of the 8^th Croatian Congress of Mathematics, Osijek, 2024.

Teaching

Diferencijalni račun (zimski semestar)

utorak (Tue), 8:00 – 12:00, D 1

Kriptografija (zimski semestar)

srijeda (Wed), 14:00 – 18:00, D 2

Integralni račun (ljetni semestar)

utorak (Tue), 8:00 – 10:00, D 1

Konzultacije (Office Hours)

Nakon održane nastave i po dogovoru.

Završni i diplomski radovi 

Teme  završnih i diplomskih radova izravno se dogovaraju na konzultacijama s nastavnikom. Dakle, o prijedlogu teme odgovarajućeg rada potrebno je kontaktirati nastavnika.