Related papers: Primes Appearing in Prime Tower Factorization
We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we…
In this short paper we present an elementary proof of the infinitude of primes. Our proof is similar in spirit to Euler's proof that the reciprocals of primes diverges and only uses tools from elementary number theory and calculus. In…
In the proposed matrix primes, through which one can readily generate a sequence of primes. The paper also proposes a number of theorems proved by which an infinite number of prime numbers twins
Simplification of fractional powers of positive rational numbers and of sums, products and powers of such numbers is taught in beginning algebra. Such numbers can often be expressed in many ways, as this article discusses in some detail.…
We adopt an empirical approach to the characterization of the distribution of twin primes within the set of primes, rather than in the set of all natural numbers. The occurrences of twin primes in any finite sequence of primes are like…
Many representation schemes combining first-order logic and probability have been proposed in recent years. Progress in unifying logical and probabilistic inference has been slower. Existing methods are mainly variants of lifted variable…
Statistical distribution of the primes in an arithmetic progression is considered. The estimation of prime numbers is given and combinatorial methods are used to calculate the twin primes on the available interval. The distribution and…
Truncated Fourier, Gauss, Kummer and exponential sums can be used to factorize numbers: for a factor these sums equal unity in absolute value, whereas they nearly vanish for any other number. We show how this factorization algorithm can…
The article describes prime intervals into the prime factorization of the middle binomial coefficient. Prime factors and prime powers are distributed in layers. Each layer consists of non-repeated prime numbers which are chosen (not…
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 a simple, closed formula which gives all the primes in order. It is a simple product of integer floor and ceiling functions.
This paper is devoted to the theory of prime numbers. In this paper we first introduce the notion of a matrix of prime numbers. Then, in order to investigate the density of prime numbers in separate rows of the matrix under consideration,…
We prove that for all $n$, simultaneously, we can choose prime filtrations of $R/I^n$ such that the set of primes appearing in these filtrations is finite.
This text investigates relations between two well-known family of algorithms, matrix factorisations and recursive linear filters, by describing a probabilistic model in which approximate inference corresponds to a matrix factorisation…
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 some observations about the well-known Goldbach conjecture. In particular we list and interpret some numerical results which allow us to formulate a relation between prime numbers and even integers. We can also…
Any rational number can be factored into a product of several rationals whose sum vanishes. This simple but nontrivial fact was suggested as a problem on a maths olympiad for high-school students. We completely solve similar questions in…
We study arithmetic properties of factorizations of elements into products of generators, in monoids given with explicit presentations. After relating and comparing this perspective to the more usual approach of factoring into products of…
We study the probability that the product of two randomly chosen elements in a finite ring $R$ is equal to some fixed element $x \in R$. We calculate this probability for semisimple rings and some special classes of local rings, and find…
Ferrers diagrams are used to visually represent integer partitions. We describe a way to use Ferrers diagrams to uniquely represent integers in terms of their prime factors. This leads to a lower bound on the number of primes less than a…