Related papers: On the 3-adic Valuation of a Cubic Binomial Sum

A new polynomial sieve is presented and used to show that almost all integers have at most one representation as a sum of two values of a given polynomial of degree at least 3.

In this paper, we investigate the distribution of the maximum of partial sums of certain cubic exponential sums, commonly known as "Birch sums". Our main theorem gives upper and lower bounds (of nearly the same order of magnitude) for the…

We give positive answer to two conjectures posed by M. E. H Ismail in his monograph [Classical and quantum orthogonal polynomials in one variable, Cambridge University Press, 2005].

An alternative computational approach to the Collatz (3n+1) conjecture is presented that may be theoretically capable of confirming the conjecture.

The vector-valued $T(1)$ theorem due to Figiel, and a certain square function estimate of Bourgain for translations of functions with a limited frequency spectrum, are two cornerstones of harmonic analysis in UMD spaces. In this paper, a…

We evaluate various binomial sums involving the powers of Fibonacci and Lucas numbers.

I. P. Goulden, S. Litsyn, and V. Shevelev [On a sequence arising in algebraic geometry, J. Integer Sequences 8 (2005), 05.4.7] conjectured that certain Laurent polynomials associated with the solution of a functional equation have only odd…

We prove an estimate for multi-variable multiplicative character sums over affine subspaces of $\mathbb A^n_k$, which generalize the well known estimates for both classical Jacobi sums and one-variable polynomial multiplicative character…

Let $r_3(n)$ be the number of representations of a positive integer $n$ as a sum of three squares of integers. We give two distinct proofs of a conjecture of Wagon concerning the asymptotic value of the mean square of $r_3(n)$.

We consider a sequence of sums of powers of the the roots of the cubic equation characterizing the Tribonacci sequences and derive its relationship with a particular Tribonacci sequence. Then we make a conjecture on the possible…

In \cite{csc}, Cetin et al. defined a new special finite sum which is denoted by $B_{1}(h,k)$. In this paper, with the help of the Hardy and Dedekind sums we will give many properties of the sum $B_{1}(h,k).$ Then we will give the…

In this note, we confirm two conjectural supercongruences on double sums of binomial coefficients due to El Bachraoui.

We prove Simon's conjecture for 3-manifolds.

A general method to express in terms of Gauss sums the number of rational points of subschemes of projective schemes over finite fields is applied to the image of the triple embedding $\mathbb{P}^1\hookrightarrow\mathbb{P}^3$. As a…

We address three questions posed by Bibak \cite{KB20}, and generalize some results of Bibak, Lehmer and K G Ramanathan on solutions of linear congruences $\sum_{i=1}^k a_i x_i \equiv b \Mod{n}$. In particular, we obtain explicit expressions…

In this paper we investigate some strong convergence theorems for partial sums with respect to Vilenkin system.

We establish the multiplicity conjecture of Herzog, Huneke, and Srinivasan about the multiplicity of graded Cohen-Macaulay algebras over a field, for codimension two algebras and for Gorenstein algebras of codimension three. In fact, we…

We prove the analogue of an identity of Huard, Ou, Spearman and Williams and apply it to evaluate a variety of sums involving divisor functions in two variables. It turns out that these sums count representations of positive integers…

We show that almost every positive integer can be expressed as a sum of four squares of integers represented as the sums of three positive cubes.

In this paper will be proved an inequality regarding $v_2(a^{b}-c^{d})$. Using this formula it will be possible to have informations about the divisibility of 2 of this function without computing it. Then, will be studied the behavior of…