Related papers: Finding Large Primes
A nonempty finite set of positive integers A is relatively prime if gcd(A) = 1 and it is relatively prime to n if gcd(A [ fng) = 1. The number of nonempty subsets of A which are relatively prime to n is \Phi(A, n) and the number of such…
Let $d\ge 3$ be a fixed integer, $p\in (0,1)$, and let $n\geq 1$ be a positive integer such that $dn$ is even. Let $\mathbb{G}(n, d, p)$ be a (random) graph on $n$ vertices obtained by drawing uniformly at random a $d$-regular (simple)…
In this paper, after presenting the results of the generalization of Pascal triangle (using powers of base numbers), we examine some properties of the 112-based triangle, most of all regarding to prime numbers. Additionally, an effective…
In this article, we will use elementary number theory techniques to investigate a sequence of integers defined by a sifting process called the lucky numbers. Ulam introduced lucky numbers as a sieve-based analogue of prime numbers. We…
The following work is written in easy language for college level students. It shows how the first digit probabilities of a group of continuous real-valued functions can be calculated. Thus, examples explaining how the probabilities are…
This article presents an efficient algorithm to generate a discrete uniform distribution on a set of $p$ elements using a biased random source for $p$ prime. The algorithm generalizes Von Neumann's method and improves computational…
Motivated by Euler's observation that the polynomial $x^{2} + x + 41$ takes on prime values for $0 \leq x \leq 39$, we search for large values of $x$ for which $N = x^{2} + x + 41$ is prime. To apply classical primality proving results…
The set of prime numbers has been analyzed, based on their algebraic and arithmetical structure. Here by obtaining a sort of linear formula for the set of prime numbers, they are redefined and identified; under a systematic procedure it has…
The purpose of this note is to report on the discovery of the primes of the form $p=1+n!\sum n$, for some natural numbers $n>0$. The number of digits in the prime p are approximately equal to $\lfloor log_{10}(1+n!\sum n)\rceil+1$.
We identify pairs of positive integers $(t, d)$ with the property that the integer sequence with general term $\lfloor{n^t/d\rfloor}$ contains at most finitely many primes.
We call an integer N>1 primover to base a if it either prime or overpseudoprime to base a. We prove, in particular, that every Fermat number is primover to base 2. We also indicate a simple process of receiving of primover divisors of…
Translated from the Latin original "Facillima methodus plurimos numeros primos praemagnos inveniendi" (1778). E718 in the Enestrom index. If m is a number of the form 4k+1 and is a sum of two relatively prime squares, then it is prime if…
In this paper, we present an improvement for the problem of deterministically finding an element of large multiplicative order modulo some integer $N$. This problem arises as a key subroutine in current deterministic factoring algorithms,…
We give a new sufficient condition which allows to test primality of Fermat's numbers. This characterization uses uniquely values at most equal to tested Fermat number. The robustness of this result is due to a strict use of elementary…
Let $k\ge 1$ be an integer, and let $P= (f_1(x), \ldots, f_k(x) )$ be $k$ admissible linear polynomials over the integers, or \textit{the pattern}. We present two algorithms that find all integers $x$ where $\max{ \{f_i(x) \} } \le n$ and…
We consider the problem of finding small prime gaps in various sets of integers $\mathcal{C}$. Following the work of Goldston-Pintz-Yildirim, we will consider collections of natural numbers that are well-controlled in arithmetic…
The notion of chiral prime concatenations is studied as a recursive construction of prime numbers starting from a seed set and with appropriate blocks to define the primality growth, generation by generation, either from the right or from…
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…
We propose a criterion that allows one to distinguish prime numbers from compound ones. This criterion is based on the counting of small quadratic residues.
Currently there is no known efficient formula for primes. Besides that, prime numbers have great importance in e.g., information technology such as public-key cryptography, and their position and possible or impossible functional generation…