### Ivan Soldo

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

- M. Jukić Bokun, I. Soldo, Extensions of D(-1)-pairs in some imaginary quadratic fields, New York Journal of Mathematics (2024), prihvaćen za objavljivanjeIn 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.
- 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 $a<2^i p^j$. 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 $a>1$. - Y. Fujita, I. Soldo, The non-existence of $D(-1)$-quadruples extending certain pairs in imaginary quadratic rings, Acta Mathematica Hungarica
**170**/2 (2023), 455-482A $D(n)$-$m$-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 $q$. - Y. Fujita, I. Soldo, D(−1)-tuples in the ring Z[√−k] with k > 0, Publicationes Mathematicae
**100**(2022), 49-67Let 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. - 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), 292-302Let $p$ 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,
*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*, XXXJournées Arithmétiques, July 3-7, 2017, Caen, France.^{th} - 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*, Conference on Diophantine**Z**[√-t], t>0*m*-tuples and related problems, November 13-15, 2014, Purdue University North Central, Westville, Indiana, USA. - I. Soldo,
*D(z)-quadruples in the ring*, Erdös Centennial**Z**[√-2], for some exceptional cases o z*,*July 1-5, 2013, Budapest, Hungary. - I. Soldo,
*The problem of existence of Diophantine quadruples in***Z**[√-2],*,*June 18 – 21, 2012, Rijeka, Croatia*.* - I. Soldo,
*Diophantine quadruples in*, Number Theory and Its Applications, An International Conference Dedicated to Kálman Győry, Attila Pethő, János Pintz and András Sárközy, Debrecen, Hungary, 2010.**Z**[√-2] - 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.