There is an absolute constant $D_0> 0$ such that if $f(x)$ is an integer polynomial, then there is an integer $\lambda$ with $|\lambda | \leq D_0$ such that $x^n +f(x) +\lambda$ is irreducible over the rationals for infinitely many integers $n\geq 1$. Furthermore, if $\deg f \leq 25$, then there is a $\lambda$ with $\lambda\in \{-2,-1,0,1,2,3\}$ such that $x^n +f(x) +\lambda$ is irreducible over the rationals for infinitely many integers $n\geq 1$. These problems arise in connection with an irreducibility theorem of Andrzej Schinzel associated with coverings of integers and an irreducibility conjecture of Pál Turán.

For a rational valued periodic function, we associate a Dirichlet series and provide a new necessary and sufficient condition for the vanishing of this Dirichlet series specialized at positive integers. This question was initiated by Chowla and implemented by Okada for a particular infinite sum. Our approach relies on the decomposition of the Dirichlet characters in terms of primitive characters. Using this, we find some new family of natural numbers for which a conjecture of Erdős holds.

The notion of $\theta$-congruent number is a generalization of congruent number, where one considers the area of a triangle with all possible angles $\theta$ such that $\cos \theta$ is rational rather than just $\theta=\frac{\pi}{2}.$ In this talk we provide a criterion for a natural number to be a $\theta$-congruent number over certain classes of real number fields.

A set of positive integers $\{a_1, a_2, ... , a_m\}$ is said to be a Diophantine m-tuple if $a_i a_j +1$ is a perfect square for all distinct $i$ and $j$. A natural question is how large can a Diophantine tuple be. In this context, the folklore Diophantine quintuple conjecture, recently settled by He, Togbe and Ziegler, states that there are no Diophantine quintuples. In this talk, we will discuss a generalization of this problem to k-th powers. This is joint work with Ram Murty and Seoyoung Kim

The Lucas sequences $U_n(r,s)$ and $V_n(r,s)$ are integers sequences that satisfy the recurrence relation \begin{equation*} %\label{eq:2} U_{n+2}= ru_{n+1}-su_{n} \end{equation*} Where $r$ and $s$ are fixed positif integers.\medskip More generally, Lucas sequences $U_n(r,s)$ and $V_n(r,s)$ represents sequence of polynomial $r$ and $s$ with integer coefficients.

Let $\alpha$ and $\beta $ denote the two roots of the equation \begin{equation*} %\label{eq:2} x^2 -rx -s = 0 \end{equation*} It has a Discriminant $ \Delta = r^2-4s$ and then the roots are $$ \alpha=\frac{r + \sqrt{\Delta}}{2} and \beta=\frac{r - \surd\Delta}{2}. $$ Thus $$\alpha + \beta = r, \alpha\beta= s, \alpha - \beta = \sqrt{\Delta}$$ and, \begin{equation*} \label{eq:2} U_{n}= a\alpha^n +b\beta^n ~~for~~ all~~ n\geq 0 . \end{equation*} where $a$ and $b$ are two constants which can be determined. The binary recurrence sequence $(u_n)_{n\geq0}$ is called nondegenerate if $ab \neq0$ and $\alpha/\beta$ is not a root of unity. \medskip Given an integer $g>1$, a base $g$-repdigit is a number of the form $$N=a\cdot\frac{g^m-1}{g-1} \quad\quad \hbox{for some $m\geq 1$ and $a\in\{1,\ldots,(g-1)\}$}.$$ When $g=10$, such number are better know as a repdigit. Recently, investigation of the repdigits in the second-order linear recurrence sequences has been of interest to mathematicians. In this talk, we will make a survey of the recents results obtained on this subject.

A quadratic form over a totally real number field $K$ is universal if it represents all totally positive integers in $K$. I'll start with a brief overview of the known results and main tools for studying universal forms, which prominently include continued fractions and good rational approximations. Then I'll explain our new result saying that: For every fixed positive integer $r$, real quadratic fields $K$ that admit a universal quadratic form of rank $r$ have density $0$. Joint work with Dayoon Park, Pavlo Yatsyna, and Blazej Zmija.

Let $\zeta_K(s)$ denote the Dedekind zeta function of a number field $K$ and $\zeta_K'(s)$ denote its derivative. In this talk, we will discuss the non-vanishing of $\zeta_K(1/2)$ and $\zeta_K'(1/2)$ and their interrelation. We have a satisfactory answer for lower degree number fields. For abelian extensions, we improve a result of Murty and Tanabe, both qualitatively and quantitatively. We also extend our investigation to Galois as well as arbitrary number fields, borrowing tools from algebraic as well as transcendental number theory.

A number field $K$, with ring of integers $O_K$, is said to be a Pólya field if the $O_K$-module formed by the ring of integer-valued polynomials on $O_K$ admits a regular basis. The Pólya group $Po(K)$ of $K$ is a particular subgroup of the ideal class group $cl(K)$ of $K$, that measures the failure of $K$ being a Pólya field. In this talk we discuss a new family of quintic non-Pólya fields associated to Lehmer quntics. It is an interesting problem to study the embedding of a number field in a Pólya field. For this family, We will also explore bounds on the degree of smallest Pólya fields containing them. Finally we show that such non-Pólya fields are non-monogenic number fields. This is a joint work with Prem Prakash Pandey.

For a given finitely generated multiplicative subgroup of the rationals which possibly contain negative numbers, we derive the densities of primes for which the index of the reduction group has a given value (under GRH). Likewise, we completely classify, in the case of rank one, torsion groups for which the density vanishes, moreover we prove that the set of primes for which the index of the reduction group has a given value, is finite. For higher rank groups we propose some partial results. Furthermore, we compute the density of the set of primes for which the order of the reduction group is divisible by a given integer.

The notion of classical visibility from the origin has been generalized by viewing lattice points through curved lines of sights. In this talk, I will generalize the notion of visible lattice points for any polynomial family of curves passing through the origin. I will show that except for the family of curves represented by monomials $x^k$, the density of visible lattice points for any other polynomial family of curves passing through the origin is always one. This is joint work with Sneha Chaubey and Shvo Regavim.

In this short talk, we will discuss the question of linear independence of special values of Dirichlet L-functions. This problem is intimately connected to relations among the special values of the Hurwitz zeta-function as well as the polylogarithms. We will mention recent progress in this context, in joint works with Ram Murty and Abhishek Bharadwaj.

While the abc conjecture remains open, much work has been done on weaker versions, and on generalising the conjecture to number fields. Stewart and Yu were able to give an exponential bound for $\max \{a, b, c\}$ in terms of the radical over the integers, while Győry was able to give an exponential bound for the projective height $H (a, b, c)$ in terms of the radical for algebraic integers. We generalise Stewart and Yu's method to give an improvement on Győry's bound for algebraic integers, before briefly discussing applications to the effective Skolem-Mahler-Lech problem and the $XY Z$ conjecture.

In 1967 Hooley proved (under GRH) Artin's conjecture on primitive roots: for any $g\in\mathbb Z\setminus\{-1,0,1\}$ which is not a square, there are infinitely many primes $p$ such that $g$ is a primitive root modulo $p$ (i.e. the multiplicative order of $(g \bmod p)$ equals $p-1$). Several variations of this problem have been studied since then. We consider the condition that the multiplicative order of $(g \bmod p)$ is divisible by a given integer, or more generally that it lies in a given arithmetic progression. We address such questions for number fields and the reductions of algebraic numbers.