Related papers: Note on the additive complements of primes

We prove some theorems which give sufficient conditions for the existence of prime numbers among the terms of a sequence which has pairwise relatively prime terms.

In [Tha15], we looked at two (`multiplicative' and `Carlitz-Drinfeld additive') analogs each, for the well-known basic congruences of Fermat and Wilson, in the case of polynomials over finite fields. When we look at them modulo higher…

We study a special set of constellations of primes generated by twin primes.

A recent heuristic argument based on basic concepts in spectral analysis showed that the twin prime conjecture and a few other related primes counting problems are valid. A rigorous version of the spectral method, and a proof for the…

We expand the theoretical background of the recently introduced superadditive and subadditive transformations of aggregation functions $A$. Necessary and sufficient conditions ensuring that a transformation of a proper aggregation function…

In this paper we prove some transformation formulae for congruences modulo a prime and deduce some congruences for Domb numbers and Almkvist-Zudilin numbers. We also pose some conjectures on congruences modulo prime powers.

The undecidability of the additive theory of primes (with identity) as well as the theory Th(N,+, n -> p\_n), where p\_n denotes the (n+1)-th prime, are open questions. As a possible approach, we extend the latter theory by adding some…

Starting with Zhang's theorem on the infinitude of prime doubles, we give an inductive argument that there exists an infinite number of prime $k$-tuples for at least one admissible set $\mathcal{H}_k=\{h_1,\ldots,h_k\}$ for each $k$.

We consider the representation of primes as a sum of a prime and twice a triangular number. We prove that a subset of the primes having density 1 is expressible in this form. We conjecture that every odd prime number is expressible as a sum…

Theory of $n$-complements with applications is presented.

An asymptotic formula for the sum of the first n primes is derived. This result improves the previous results of P. Dusart. Using this asymptotic expansion, we prove the conjectures of R. Mandl and G. Robin on the upper and the lower bound…

We resolve the function field analogue of the conjecture concerning distribution of twin primes in arithmetic progression.

In this paper, we introduce the method of adding additional factors and a parameter to multiple zeta values and prove some generalizations of the duality theorem and several relations among multiple zeta values. In particular, we are able…

This is the second of two coupled papers estimating the mean values of multiplicative functions, of unknown support, on arithmetic progressions with large differences. Applications are made to the study of primes in arithmetic progression…

We proved that there are infinitely many pairs of twin prime.

We prove an asymptotic formula for the number of primes of the shape $a^2 +p^4$, thereby refining the well known work of Friedlander and Iwaniec. Along the way, we prove a result on equidistribution of primes up to $x$, in which the moduli…

Every natural number greater than two may be written as the sum of a prime and a square-free number. We establish several generalisations of this, by placing divisibility conditions on the square-free number.

Suppose that $n$ is $0$ or $4$ modulo $6$. We show that there are infinitely many primes of the form $p^2 + nq^2$ with both $p$ and $q$ prime, and obtain an asymptotic for their number. In particular, when $n = 4$ we verify the `Gaussian…

Ruzsa's equivalence theorem provided a framework for converting certain families of inequalities in additive combinatorics to entropic inequalities (which sometimes did not possess stand-alone entropic proofs). In this work, we first…

In this paper, we establish some theorems on the distribution of primes in higher-order progressions on average.