Related papers: On an average ternary problem with prime powers

We prove two estimates for averages of sums of Kloosterman fractions over primes. The first of these improves previous results of Fouvry-Shparlinski and Baker.

In this paper, we pose lots of challenging conjectures on congruences for the sums involving binomial coefficients and Ap\'ery-like numbers modulo $p^3$, where $p$ is an odd prime.

This work proposes a proof of the simplest cubic primes counting problem. It shows that the subset of primes {p = n^3 + 2 is prime : n => 1} is an infinite subset of primes. Further, the expected order of magnitude of the cubic primes…

We study, from the viewpoint of metrical number theory and (infinite) ergodic theory, the probabilistic laws governing the occurrence of prime numbers as digits in continued fraction expansions of real numbers.

In this paper, we present several explicit formulas of the sums and hyper-sums of the powers of the first (n+1)-terms of a general arithmetic sequence in terms of Stirling numbers and generalized Bernoulli polynomials.

We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.

We offer some partition functions related to ternary quadratic forms, and note on their upper bounds and related properties. We offer these results as an application of a simple method related to conjugate Bailey pairs presented in a prior…

We survey some recent developments and give a list of open problems regarding multiple recurrence and convergence phenomena of $\mathbb{Z}^d$ actions in ergodic theory and related applications in combinatorics and number theory.

The ternary Goldbach conjecture (or three-prime conjecture) states that every odd number greater than 5 can be written as the sum of three primes. The purpose of this book is to give the first proof of the conjecture, in full.

In the paper I considered algebra of polynomials over associative D-algebra with unit. Using the tensor notation allows to simplify the representation of polynomial. I considered questions related to divisibility of polynomial of any power…

We present a new sieve that allows us to find the prime numbers by using only regular patterns and, more importantly, avoiding any duplication of elements between them.

Following an idea of Rowland we give a conjectural way to generate increasing sequences of primes using algorithms involving the gcd. These algorithms seem not so useless for searching primes since it appears we found sometime primes much…

In this work we consider sums of primes that converging very slow. We set as a base, a reformulation of analytic prime number theorem and we use the values of Riemann Zeta function for the approximation. We also give the truncation error of…

We introduce and consider a certain probability question involving elementary number theory and the likelihood that a fixed prime will appear in a certain recursively defined factorization of an integer. We derive several convergent…

We prove an explicit formula for the $p$-adic valuation of the Legendre polynomials $P_n(x)$ evaluated at a prime $p$, and generalize an old conjecture of the third author. We also solve a problem proposed by Cigler in 2017.

For A,epsilon>0 and any sufficiently large odd n we show that for almost all k up to n^{1/5-epsilon} there exists a representation n=p1+p2+p3 with primes in residue classes b1,b2,b3 mod k for almost all admissible triplets b1,b2,b3 of…

Extending a classical estimate of Mertens for the sum of the reciprocals of the first primes, we provide an explicit remainder formula for products of an arbitrary, but fixed, number of primes.

Let $\{u_{n}\}_{n \geq 0}$ be a non-degenerate binary recurrence sequence with positive, square-free discriminant and $p$ be a fixed prime number. In this paper, we have shown the finiteness result for the solutions of the Diophantine…

We present and analyze an algorithm to enumerate all integers $n\le x$ that can be written as the sum of consecutive $k$th powers of primes, for $k>1$. We show that the number of such integers $n$ is asymptotically bounded by a constant…

We give a broad survey of recent results in Enumerative Combinatorics and their complexity aspects.