## Talks and Posters

The announcement of the conference schedule is scheduled for August 17th, 2022.

**Higher Reciprocity Law and Linear Recurrence Sequences**Armand Noubissie - Max Planck Institute for Software Systems - Germany

In this talk, we describe the set of prime numbers splitting completely in the non-abelian splitting field of certain monic irreducible polynomials of degree $3.$ As an application we establish some divisibility properties of the associated ternary recurrence sequence by primes $p$, thus greatly extending recent work of Evink and Helminck and of Faisant. We also prove some new results on the number of solutions of the characteristic equation of the recurrence sequence modulo $p,$ extending and simplifying earlier work of Zhi-Hong Sun (2003). This is joint work with Pieter Moree.

**$m$-universal quadratic forms in real quadratic fields**Błażej Żmija - Charles University in Prague - Czech Republic

Let $m$-be a positive integer, $D$ be a squarefree positive integer and let $Q$ be a positive definite quadratic form with the coefficients in $O_{K}$, where $K=\mathbb{Q}(\sqrt{D})$. We say that $Q$ is $m$-universal if it represents all totally positive elements of $mO_{K}$.

During my talk, I will present results regarding the minimal number of variables of any $m$-universal quadratic form in $\mathbb{Q}(\sqrt{D})$. In particular, I will explain why for every fixed $B$, the set all all numbers $D$ such that there exists an $m$-universal quadratic form in $\mathbb{Q}(\sqrt{D})$ in at most $B$ variables, has natural density zero. The talk is based on joint work with V. Kala, D. Park and P. Yatsyna.

**The link between Hopf-Galois theory and skew braces**Daniel Gil-Muñoz - Charles University in Prague - Czech Republic

The group ring of the Galois group on a finite Galois extension of fields is a Hopf algebra; one may then define a Hopf-Galois structure on an extension as a Hopf algebra together with a linear action satisfying properties in the fashion of a Galois group action. This gives rise to the so-called Hopf-Galois theory, which was introduced in the sixties of last century and has been found to be a fruitful generalization of Galois theory. In 2016, it was discovered a correspondence between Hopf-Galois structures and skew braces. A skew brace is a set endowed with two different group laws and satisfying a twisted distributive property. Since its appearance, the theory of skew braces has been of special interest due to its usefulness to construct set-theoretic solutions of the Yang-Baxter equation. In this talk we shall give a comprehensive introduction to both theories and discuss some of their intersections.

**A Fibonacci type sequence with Prouhet-Thue-Morse coefficients**

Eryk Tadeusz Lipka - Pedagogical University in Krakow - Poland

Let \(t_{n}=(-1)^{s_{2}(n)}\), where \(s_{2}(n)\) is the sum of binary digits function. The sequence \((t_{n})_{n\in\mathbb{N}}\) is the well-known Prouhet-Thue-Morse sequence. We will consider a sequence \((h_n)_{n \in\mathbb{N}}\) defined by \(h_0 = 0, h_1=1\) and \(h_n=t_nh_{n-1}+h_{n-2}\) for \(n\ge2\). As this sequence is similar to Fibonacci sequence we expect it to have some similar properties. We will prove several results concerning arithmetic properties of the sequence \((h_{n})_{n\in\mathbb{N}}\). In particular, we prove non-vanishing for \(n\ge 5\), and automaticity of the sequence \((h_{n}\pmod{m})_{n\in\mathbb{N}}\) for each \(m \in \mathbb{N}_{\ge2}\). We will also discuss whether some of those properties hold when \((t_n)_{n\in \mathbb N}\) is replaced with other automatic sequence. Talk based on recent paper with Maciej Ulas.

**Signs behaviour of sums of weighted numbers of partitions**

Filip Gawron - Jagiellonian University in Krakow - Poland

Let $A$ be a subset of positive integers. By $A$-partition of $n$ we understand the representation of $n$ as a sum of elements from the set $A$. For given $i$, $n\in \mathbb{N}$, by $c_A(i,n)$ we denote the number of $A$-partitions of $n$ with exactly $i$ parts. In the talk I will describe several results concerning sign behaviour of the sequence $S_{A,k}(n) = \sum_{i=0}^n(-1)^i i^k c_A(i, n)$, for fixed $k\in \mathbb{N}$. Especially I will focus on the periodicity of the sequence of signs for different forms of $A$. Finally, I will also mention some conjectures and questions that arose naturally during our research.The talk is based on a joint work with Maciej Ulas (Jagiellonian University).

**A study on the use of Pell hyperbolas in DLP-based cryptosystems**

Gessica Alecci - Politecnico di Torino - Italy

The group structure of the Pell hyperbolas finds numerous applications in various cryptosystems and in particular in Public-Key Encryption (PKE) schemes whose security is based on DLP, such as the ElGamal PKE scheme.

In this talk, we will show a generalization of the group structure on Pell hyperbolas and provide a parameterization from both an algebraic approach and a geometrical interpretation. Moreover, we will introduce some novel ElGamal schemes over the Pell hyperbola discussing their advantages from a computational point of view thanks to a particular parametrization.

**Real quadratic fields with large class number**Giacomo Cherubini - Charles University in Prague - Czech Republic

We know that any integer can be factored in a unique way as a product of primes. This is no longer true over number fields and the class number indicates "how badly unique factorization fails": if the class number is one then we have unique factorization, while anything bigger than one means we don't. A long-standing open conjecture of Gauss states that there are infinitely many real quadratic fields with class number one. In the opposite direction, one can prove that there are infinitely many real quadratic fields with class number as large as possible. In this talk I will explain what "as large as possible" means and a few ideas on how the result can be proved.

**From the Bessenrodt-Ono type inequality to the $r$-log-concavity of the restricted partition function $p_\mathcal{A}(n,k)$**

Krystian Gajdzica - Jagiellonian University in Krakow - Poland

Let $\mathcal{A}=\left(a_i\right)_{i=1}^\infty$ be a non-decreasing sequence of positive integers, and let $k\in\mathbb{N}_+$ be fixed. For an arbitrary integer $n\geqslant0$, the restricted partition function $p_\mathcal{A}(n,k)$ is the number of partitions of $n$ with parts in the multiset $\{a_1,a_2,\ldots,a_k\}$.

At the beginning of the talk, we will investigate under what conditions on $\mathcal{A}$ and $k$ the Bessenrodt-Ono type inequality:

$p_\mathcal{A}(a,k)p_\mathcal{A}(b,k)>p_\mathcal{A}(a+b,k)$

holds for all sufficiently large values of $a$ and $b$. Afterward, the so-called log-behaviour of the function $p_\mathcal{A}(n,k)$ will be considered in the same manner. Among other things, we will examine when the sequence $\left(p_\mathcal{A}(n,k)\right)_{n=0}^\infty$ is log-concave --- the inequality

$p_\mathcal{A}^2(n,k)>p_\mathcal{A}(n+1,k)p_\mathcal{A}(n-1,k)$

is satisfied for every large parameter $n$. Additionally, the log-balancedness, the higher order Tur\'an inequalities and the $r$-log-concavity for the restricted partition function will be discussed.

**Riesel Numbers**

Krzysztof Kowitz - University of Gdańsk - Poland

We will introduce ``Riesel" numbers and we show closed formulae for calculating the $n$th ``Riesel" number $R_n$, the number of ``Riesel" numbers up to $n$, and the sum of the first $n$ ``Riesel" numbers. We will examine the relationship between some known numbers.

**Brun's sieve**

Łukasz Orski - Jagiellonian University in Krakow - Poland

The first known example of a sieve is the sieve of Eratosthenes. However, from ancient times there was no bigger results in the sieve theory up until the twentieth century. In 1915 Norwegian mathematician Viggo Brun modified the idea of Eratosthenes. It allowed him to estimate $B_2(x)$ - the number of twin primes $\leqslant x$. It was enough to prove that the sum of the reciprocals of the twin primes converges. Since then, the sieve theory has become one of the main branches of modern number theory. In this talk I will discuss the general idea of the sieve in mathematics and difficulties in its application. I will also show how Viggo Brun managed to avoid these difficulties and I will prove the Brun's Theorem.

**From counting holes to Riemann hypothesis**Marcin Oczko - Jagiellonian University in Krakow - Poland

Weil conjectures present a very deep connection between topology and number theory. They tell us that by counting solutions of algebraic equations over finite fields, we can explore their underlying topological structure. With the help of the Lefschetz trace formula, I will try to convey some intuition about the precise statement of Weil conjectures and present a very rough idea of their proof. During my talk I will recall some necessary properties of cohomology groups and algebraic varieties.

**Some connections between local zeta function and p-adic analysis**

Mieszko Zimny - University of Warsaw - Poland

The Gross–Koblitz formula expresses a Gauss sum using the values of the p-adic gamma function. It allows the use of p-adic analysis methods to study the local zeta function. My master's project, written under the supervision of M. Vlasenko, concerns some closely related connections between this two fields. I'm going to briefly present an overview of our work and some results.

**Number of elements of small norm in the simplest cubic fields**Mikuláš Zindulka - Charles University in Prague - Czech Republic

We work in a family of the \textbf{simplest cubic fields} $ K_a $ parametrized by one integer parameter $ a $. Our goal is to estimate the number of integral elements (up to conjugation and multiplication by units) whose norm is below a certain bound $ X $. We provide an asymptotic estimate depending on $ a $ and $ X $ up to a multiplicative constant which is independent of the two parameters.

The study of elements of small norm was initiated in a paper by Lemmermeyer and Peth\""{o}, who showed that the norm of any non-unit integral element of $ K_a $ is at least $ 2a+3 $ and that the minimum is attained. The result was further extended by Kala and Tinkov\'{a}, who gave an estimate for the number of elements with norm $ \leq a^2 $. The count is much larger than predicted by the \textbf{class number formula}.

We give a natural explanation of this discrepancy and show that if $ X \geq a^4 $, then the number of elements agrees with the heuristics coming from the class number formula.

**Elliptic Curves and Beyond! - POSTER**Saikat Maity - Pondicherry University - India

What is an Elliptic Curve?, The addition group law on Elliptic curves E(Q), Rational Points on Elliptic Curves E(Q) E(Z), Elliptic curves over finite fields, Fermats Last Theore (only result), Connection between elliptic curves and cryptography (only Elliptic Curve Diffie-Hellman Key Exchange).

**Parity bias for partitions into distinct parts**Siu Hang Man - Charles University in Prague - Czech Republic

Recently, Kim, Kim and Lovejoy proved that for $n\ge 3$, there are more partitions of $n$ with more odd parts than even parts, than vice versa; they also proved an asymptotic formula for these classes of partitions. In this talk we establish that a similar phenomenon also exists for partitions into distinct parts. As a generalisation, we show that we can actually compare the number of parts of a partition in any residue classes. This is joint work with Kathrin Bringmann, Larry Rolen, and Matthias Storzer.

**A matrix analogue of Schur-Siegel-Smyth Trace Problem**Srijonee Shabnam Chaudhury - Harish-Chandra Research Institute - India

Let $\mathcal{S}$, $\mathcal{T}$, $\mathcal{C}$ be three sets of some special types of connected positive definite matrices. The \textit{Absolute Trace} of a square matrix $A$ is defined by \frac{\text{Trace of matrix A}}{\text{Order of A}}

In this paper, we extend the notion of absolute trace by defining \textit{trace-$k$ measure} of $A$ $=$ mean value of the power sums of eigenvalues of $A$. And we obtain the greatest lower bound of this measure for any $n \times n $ matrix in $\mathcal{S} \cup \mathcal{T} \cup \mathcal{C}$. Using this we find some criteria which assure when an integer polynomial can not be a characteristic or minimal polynomial of an integer symmetric matrix. We also obtain here the smallest limit point of the set of all \textit{absolute trace-$k$ measure} of matrices in $\mathcal{S} \cup \mathcal{T} \cup \mathcal{C}$.

Finally, we exhibit that the famous result of Smyth on density of absolute trace measure of totally positive integers is also true for the set of symmetric integer connected positive definite matrices.

**Modular forms and number theory**Tymoteusz Chmiel - Jagiellonian University - Poland

Modular forms are holomorphic functions on the complex upper half-plane, and as such seem to be pretty disjoint from the realm of number theory. In fact the opposite is true: connections between modular forms and number theory are numerous and very deep. From the number of ways one can represent a positive integer as a sum of four squares to the Langlands program, probably the most ambitious project in contemporary mathematics, modular forms appear in many unexpected places and connect areas that are seemingly very distant. In my talk I will introduce the notion of a modular form, focusing on its applications to number theory but also to other topics such as elliptic curves, Lie algebras and physics.