Related papers: Three Theorems on modular sieves that suggest the …
A primorial prime is a prime number of the form $p_n\# \pm 1$ where $p_n\#$ denotes the product of all primes less than or equal to $p_{n}$, the $n$-th prime. We show that the probability along the lines of Mertens' Theorem that either…
For $n \geq 3,$ let $ p_n $ denote the $n^{\rm th}$ prime number. Let $[ \; ]$ denote the floor or greatest integer function. For a positive integer $m,$ let $\pi_2(m)$ denote the number of twin primes not exceeding $m.$ The twin prime…
A prime sieve is an algorithm that finds the primes up to a bound $n$. We say that a prime sieve is incremental, if it can quickly determine if $n+1$ is prime after having found all primes up to $n$. We say a sieve is compact if it uses…
In this paper we review the properties of families of numbers of the form $6n\pm1$, with $n$ integer (in which there are all prime numbers greater than 3 and other compound numbers with particular properties) to later use them in a new…
Most prime gaps results have been proven using tools from analytic or algebraic number theory in the last few centuries. In this paper, we would like to present some probabilistic way of proving many essential results. A major component of…
In this paper we study the integrals of fractional parts of given functions, and develop some new tools to understand the behaviour of prime differences. We demonstrate how simply some seemingly difficult conjectures related to prime…
While the prime numbers have been subject to mathematical inquiry since the ancient Greeks, the accumulated effort of understanding these numbers has - as Marcus du Sautoy recently phrased it - 'not revealed the origins of what makes the…
Odd numbers can be indexed by the map k(n)=(n-3)/2, n belonging to 2N+3. We first propose a basic primality test using this index function that was first introduced in article (8). Input size of operations is reduced which improves…
We make two algorithms that generate all prime numbers up to a given limit, they are a development of sieve of Eratosthenes algorithm, we use two formulas to achieve this development, where all the multiples of prime number 2 are eliminated…
This is an expository article to accompany my two lectures at the CDM conference. I have used this an excuse to make public two sets of notes I had lying around, and also to put together a short reader's guide to some recent joint work with…
We introduce a novel sieve for prime numbers based on detecting topological obstructions in a M\"obius-transformed rational metric space. Unlike traditional sieves which rely on divisibility, our method identifies primes as those numbers…
We studied two probabilistic models of the distribution of primes in the natural number [1].The paper considers the third probabilistic model of the distribution of primes in the natural number. The author proved that the results obtained…
It is known that there are infinitely-many prime numbers which take the form of a polynomial of degree one with integer coefficients, this is Dirichlet's theorem. We use an elementary sieving argument together with bounds on the prime…
We show that there are infinitely many primes $p$ such that $p-1$ is divisible by a square $d^2 \geq p^\theta$ for $\theta=1/2+1/2000.$ This improves the work of Matom\"aki (2009) who obtained the result for $\theta=1/2-\varepsilon$ (with…
In 1977, the first author observed a duality between the largest and smallest prime factors of integers, and established as a consequence some new results on the M\"obius function $\mu(n)$ using the Prime Number Theorem for Arithmetic…
We consider primitive divisors of terms of integer sequences defined by quadratic polynomials. Apart from some small counterexamples, when a term has a primitive divisor, that primitive divisor is unique. It seems likely that the number of…
In the first paper under this title (1977), the first author utilized a duality identity between the largest and smallest prime factors involving the Moebius function, to establish the following result as a consequence of the Prime Number…
Comparative prime number theory is the study of the {\em{discrepancies}} of distributions when we compare the number of primes in different residue classes. This work presents a list of the problems being investigated in comparative prime…
The theorem presented in this paper allows the creation of large prime numbers (of order up to o(n^2)) given a table of all primes up to n.
Using as the working hypothesis of an evaluation of the difference between primes $p_{n+1} - p_n = O(\sqrt{p_n})$ we represent in detail the proofs of Legendre's and Oppermann's conjectures.