Related papers: A note on a conjecture by Ulas on polynomial subst…

We study the shifted convolution sums associated to completely multiplicative functions taking values in $\{\pm 1\}$ and give combinatorical proofs of two recent results in the direction of Chowla's conjecture. We also determine the…

We give a lower bound for the degree of an irreducible factor of a given polynomial. This improves and generalizes the results obtained in [4, On the irreducible factors of a polynomial, Proc. Amer. Math. Soc., 148 (2020] 1429 -- 1437].

In this paper we construct a new q-Euler numbers and polynomials. By using these numbers and polynomials, we give the interesting formulae related to alternating sums of powers of consecutive q-integers following an idea due to Euler.

The aim of this note is a proof of a recent conjecture of Kellner concerning the number of distinct prime factors of a particular product of primes. The proof uses profound results from analytic number theory, such as Granville-Ramar\'{e}'s…

We give a short proof, using generating functions, for a polynomial congruence for Eulerian polynomials first proved, using arrangements of hyperplanes, by Yoshinaga and later proved, using roots of unity, by Iijima, Sasaki, Takahashi, and…

We study certain kind of polynomials associated with Lissajous curves, called Chebyshev-Lissajous polynomials. We investigate their irreducibilities over the real numbers and complex numbers, thus comfirming two conjectures proposed by…

In this paper, we survey some recent results on the Artin conjecture and discuss some aspects for the Artin conjecture.

In this paper we obtain a partial answer to a conjecture on the solvabilty of linear difference equations in quasianalytic Carleman classes.

We prove a conjecture about the minimal nonnegative solutions of algebraic Riccati equations associated with reducible singular M-matrices. The result enhances our understanding of the behaviour of doubling algorithms for finding the…

The generalized Lucas numbers are polynomials in two variables with nonnegative integer coefficients. Lucas versions of some combinatorial numbers with known formulas in terms of quotient and products of nonnegative integers have been…

We prove that seminormality of cut polytopes is equivalent to normality. This settles two conjectures regarding seminormality of cut polytopes.

We give short elementary expositions of combinatorial proofs of some variants of Euler's partitition problem that were first addressed analytically by George Andrews, and later combinatorially by others. Our methods, based on ideas from a…

We prove new results, related to the Littlewood and Mixed Littlewood conjectures in Diophantine approximation.

We present a short proof of a conjecture proposed by I. Ra\c{s}a (2017), which is an inequality involving basic Bernstein polynomials and convex functions. This proof was given in the letter to I. Ra\c{s}a (2017). The methods of our proof…

In this paper, we introduce the polynomial continued fraction, a close relative of the well-known simple continued fraction expansions which are widely used in number theory and in general. While they may not possess all the intriguing…

We give a short proof of polynomial recurrence with large intersection for additive actions of finite-dimensional vector spaces over countable fields on probability spaces, improving upon the known size and structure of the set of strong…

We prove a distribution-theoretic conjecture of Robert Coleman, thereby also obtaining an explicit description of the complete set of Euler systems for the multiplicative group over Q.

The Casas-Alvero conjecture is about interpolation polynomials. There are some partial proofs of it, but there is not any proof in the general case.In this paper we propose three.

We prove a conjecture of Johann Cigler on shifted Hankel determinants.

We establish an analogue of the Goldbach conjecture for Laurent polynomials with positive integer coefficients.