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Let $P(x) \in \mathbb{Z}[x]$ be a polynomial. We give an easy and new proof of the fact that the set of primes $p$ such that $p \mid P(n)$, for some $n \in \mathbb{Z}$, is infinite. We also get analog of this result for some special…

For a prime $p\ge 5$ let $q_0,q_1,\ldots,q_{(p-3)/2}$ be the quadratic residues modulo $p$ in increasing order. We study two $(p-3)/2$-periodic binary sequences $(d_n)$ and $(t_n)$ defined by $d_n=q_n+q_{n+1}\bmod 2$ and $t_n=1$ if…

Divisibility sequences are defined by the property that their elements divide each other whenever their indices do. The divisibility sequences that also satisfy a linear recurrence, like the Fibonacci numbers, are generated by polynomials…

In this paper we study the Fibonacci numbers and derive some interesting properties and recurrence relations. We prove some charecterizations for $F_p$, where $p$ is a prime of a certain type. We also define period of a Fibonacci sequence…

Let $(x_n)_{n\geq0}$ be a linear recurrence of order $k\geq2$ satisfying $$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$, where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In [`The quotient…

We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the {\em total degree} and the…

We consider a family of integer sequences generated by nonlinear recurrences of the second order, which have the curious property that the terms of the sequence, and integer multiples of the ratios of successive terms (which are also…

We study the set $\mathcal{L}_{F}$ of all $F$-vector spaces $L(P)$ where $P$ is monic and splits over $F$ and $L(Q)$ denotes the set of linear recurrence sequences over $F$ with characteristic polynomial $Q$. We show that $\mathcal{L}_{F}$…

Using the action of the Galois group of a normal extension of number fields, we generalize and symmetrize various fundamental statements in algebra and algebraic number theory concerning splitting types of prime ideals, factorization types…

We study the generating series associated with the degree sequence of a monomial self-map of a projective toric variety. We establish conditions under which this series has its circle of convergence as a natural boundary, and hence is a…

The degree sequence of the algebraic numbers in an algebraic linear recurrence sequence is shown to be virtually periodic. This is proved using the Skolem-Mahler-Lech theorem. It has applications to the degree sequence and the minimal…

Many famous integer sequences including the Catalan numbers and the Motzkin numbers can be expressed in the form $ConstantTermOf\left[P(x)^nQ(x)\right]$ for Laurent polynomials $Q$, and symmetric Laurent trinomials $P$. In this paper we…

We establish that the sequences formed by logarithms and by "fractional" powers of integers, as well as the sequence of prime numbers, are non-holonomic, thereby answering three open problems of Gerhold [Electronic Journal of Combinatorics…

A ternary inclusion-exclusion polynomial is a polynomial of the form \[ Q_{{p,q,r}}=\frac{(z^{pqr}-1)(z^p-1)(z^q-1)(z^r-1)} {(z^{pq}-1)(z^{qr}-1)(z^{rp}-1)(z-1)}, \] where $p$, $q$, and $r$ are integers $\ge3$ and relatively prime in pairs.…

Exponential sums with monomials are highly related to many interesting problems in number theory and well studied by many literatures. In this paper, we consider the exponential sums with polynomials and prove a new upper bound. As an…

We identify a structural pattern in the construction of known infinite families of trees whose independence polynomials are not log-concave. Using this pattern and properties of polynomial ring ideals, we derive linear recurrences for these…

We study two polynomial counting questions in arithmetic statistics via a combination of Fourier analytic and arithmetic methods. First, we obtain new quantitative forms of Hilbert's Irreducibility Theorem for degree $n$ polynomials $f$…

We give bounds for the number and the size of the primes $p$ such that a reduction modulo $p$ of a system of multivariate polynomials over the integers with a finite number $T$ of complex zeros, does not have exactly $T$ zeros over the…

We generalize the polynomial Szemer\'{e}di theorem to intersective polynomials over the ring of integers of an algebraic number field, by which we mean polynomials having a common root modulo every ideal. This leads to the existence of new…

Let $p$ be a prime. In this paper, we give a complete classification of self-reciprocal polynomials arising from Fibonacci polynomials over $\mathbb{Z}$ and $\mathbb{Z}_p$, where $p=2$ and $p>5$. We also present some partial results when…