Associate Professor

### Ivan Soldo

Head of Department of Pure Mathematics and Mathematics Teaching
isoldo@mathos.hr
+385-31-224-822
19 (1st floor)
School of Applied Mathematics and Informatics

Josip Juraj Strossmayer University of Osijek

### 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. M. Jukić Bokun, I. Soldo, Extensions of D(-1)-pairs in some imaginary quadratic fields, New York Journal of Mathematics (2024), prihvaćen za objavljivanje
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. 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)
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 $a&lt;2^i p^j$. 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 $a&gt;1$.
3. 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
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 $a&lt;b&lt; 16a^2-a-2+2sqrt{k(8a^2+3a+1)}$ and integers $c$ and $d$ satisfying $d&lt;0&lt;c$. 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 $a&lt;ble 8a-3$. 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 $q$.
4. Y. Fujita, I. Soldo, D(−1)-tuples in the ring Z[√−k] with k > 0, Publicationes Mathematicae 100 (2022), 49-67
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.
5. 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
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 $Z[sqrt{-p}]$.

### 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 IIISeptember 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, 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 equationsFebruary 15-19, 2016Salzburg, Austria.
• I. Soldo, D(-1)-triples of the form {1,b,c} in the ring Z[√-t], t>0, Workshop on Number Theory and AlgebraDepartment 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 CentennialJuly 1-5, 2013, Budapest, Hungary.
• I. Soldo, The problem of existence of Diophantine quadruples in Z[√-2],  5th Croatian Mathematical CongressJune 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 the Organize Committee of the 4th Croatian Congress of Mathematics, Osijek, 2008

Member of the Organize Committee of the 15th International Conference on Operational Research, Croatian Operational Research Society, Osijek 2014

Member of the Organize Committee of the 16th 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 8th 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.