Related papers: There are infinitely many prime numbers in all ari…

Suppose that $\alpha,\beta\in\mathbb{R}$. Let $\alpha\geqslant1$ and $c$ be a real number in the range $1<c< 12/11$. In this paper, it is proved that there exist infinitely many primes in the generalized Piatetski--Shapiro sequence, which…

B. Green and T. Tao have recently proved that 'the set of primes contains arbitrary long arithmetic progressions', answering to an old question with a remarkably simple formulation. The proof does not use any "transcendental" method and any…

We give a new proof that there are infinitely many primes, relying on van der Waerden's theorem for coloring the integers, and Fermat's theorem that there cannot be four squares in an arithmetic progression. We go on to discuss where else…

In this short note, we give two proofs of the infinitude of primes via valuation theory and give a new proof of the divergence of the sum of prime reciprocals by Roth's theorem and Euler-Legendre's theorem for arithmetic progressions.

In this note we generalise a method of Perott to give new proofs that there are infinitely many prime numbers.

The Riemann zeta function, and more generally the L-functions of Dirichlet characters, are among the central objects of study in number theory. We report on a project to formalize the theory of these objects in Lean's "Mathlib" library,…

We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length $k$ in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes.

We study the number of primes with a given primitive root and in an arithmetic progression under the assumption of a suitable form of the generalized Riemann Hypothesis. Previous work of Lenstra, Moree and Stevenhagen has given asymptotics…

We give the converse to Dirichlet's theorem on primes in arithmetic progressions by generalizing an old result of Guinand.

We give new characterizations of the Midy's property and using these results we obtain a new proof of a special case of the Dirichlet's theorem about primes in arithmetic progression.

Under the assumption of Heath-Brown's conjecture on the first prime in an arithmetic progression, we prove that there are infinitely many Carmichael numbers $n$ such that the number of prime factors of $n$ is prime.

If $a$ and $d$ are relatively prime, we refer to the set of integers congruent to $a$ mod $d$ as an `eligible' arithmetic progression. A theorem of Dirichlet says that every eligible arithmetic progression contains infinitely many primes;…

This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic…

In this paper we use Dirichlet's theorem in order to elementally prove two theorems. The first says that since a polynomial ax+b generates one prime, it also generates infinites. The second theorem (which is proved in a very simillar way to…

We give an axiomatic characterization of multiple Dirichlet series over the function field $\mathbb F_q(T)$, generalizing a set of axioms given by Diaconu and Pasol. The key axiom, relating the coefficients at prime powers to sums of the…

This paper proves the existence of infinitely many Perrin pseudoprimes, as conjectured by Adams and Shanks in 1982. The theorem proven covers a general class of pseudoprimes based on recurrence sequences. The result uses ingredients of the…

In 1845, Bertrand conjectured that twice any prime strictly exceeds the next prime. Tchebichef proved Bertrand's postulate in 1850. In 1934, Ishikawa proved a stronger result: the sum of any two consecutive primes strictly exceeds the next…

A classical result in number theory is Dirichlet's theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly k prime factors for k>1. Building upon a proof by E.M. Wright in 1954, we…

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that…

The usual product $m\cdot n$ on $\mathbb{Z}$ can be viewed as the sum of $n$ terms of an arithmetic progression whose first term is $a_{1}=m-n+1$ and whose difference is $d=2$. Generalizing this idea, we define new similar product mappings,…