Related papers: Infinitely Many Primes Using Generating Functions

Infinite antichains of permutations have long been used to construct interesting permutation classes and counterexamples. We prove the existence and detail the construction of infinite antichains with arbitrarily large growth rates. As a…

A new derivation of Golomb's limit formula for generating the $n$th$+1$ prime number is presented. The limit formula is derived by extracting $p_{n+1}$ from Euler's prime product representation of the Riemann zeta function $\zeta(s)$ in the…

It is known that the sum of the reciprocal of integers, $\sum_n (1/n)$, and the sum of the reciprocal of primes, $\sum_n (1/p_n)$, both diverge. Here, we study a series made from primes that sums exactly to 1. We also show this sum is…

Let $\Omega(n)$ denote the number of prime factors of $n$. We show that for any bounded $f\colon\mathbb{N}\to\mathbb{C}$ one has \[ \frac{1}{N}\sum_{n=1}^N\, f(\Omega(n)+1)=\frac{1}{N}\sum_{n=1}^N\, f(\Omega(n))+\mathrm{o}_{N\to\infty}(1).…

In this paper we use computational method based on operational point of view to prove a new generating function of exponential polynomials. We give its applications involving geometric polynomials, Bernoulli and Euler numbers.

Let G be the space of generating functions of a periodic infinite order linear recurrence. In this paper we provide an explicit procedure for computing a basis of G.

The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function $\zeta(s)\equiv\sum_{n=1}^\infty…

We study the distribution of primes from a topological viewpoint. Certain conjecture is introduced, and we show that it is equivalent to the Riemann Hypothesis.

Numbers (positive integers) are the most fundamental creatures of the human mind and the foundation to the scientific understanding of nature. Some mathematicians have suspected a link between prime numbers and secrets of creation.…

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,…

We present in this work a heuristic expression for the density of prime numbers. Our expression leads to results which possesses approximately the same precision of the Riemann's function in the domain that goes from 2 to 1010 at least.…

Two well known facts from elementary number theory are proven by using Bergman spaces.

We present a novel approach to the Twin Prime Conjecture, basing on the $6x \pm 1$ representation of primes. By defining so-called twin prime generators $x \in \N$, for which both $6x - 1$ and $6x + 1$ are prime, we reformulate the…

We prove the theorems which are equivalent to the Roland's results such that a new form of them allows to consider some generalizations. In particular, we give generators of primes more than a fixed prime.

We prove the analog of Cram\'er's short intervals theorem for primes in arithmetic progressions and prime ideals, under the relevant Riemann Hypothesis. Both results are uniform in the data of the underlying structure. Our approach is based…

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…

Green and Tao proved that the primes contains arbitrarily long arithmetic progressions. We show that, essentially the same proof leads to the following result: The primes in an short interval contains many arithmetic progressions of any…

We study the surprising discrepancy between the number of primes corresponding, respectively, to the two letters of an infinite word engendered by one of the simplest Lindenmayer systems. We formulate a conjecture concerning the rate of…

In a recent joint work with D.A. Goldston and C.Y. Yildirim we just missed by a hairbreadth a proof that bounded gaps between primes occur infinitely often. In the present work it is shown that adding to the primes a much thinner set,…

We present an elementary inductive proof which Euler could have obtained, for the corresponding result as the title indicates, had he refined a bit his proof for Fermat's assertion on representing primes as two squares.