Related papers: Finding Large Primes
In this paper, we introduce a novel quantum algorithm for the factorization of composite odd numbers. This work makes two significant contributions. First, we present a new improvement to the classical Fermat method, fourfold reducing the…
Given a graph $G=(V,E)$ with $V=\{1,2,...,k\}$, the $k$ positive integers $a_1,a_2, ...,a_k$ are $G$-wise relatively prime if $(a_i, a_j)=1$ for $\{i,j\} \in E$. In this note we consider the problem of finding the probability $A_G$ that k…
Starting with a set of weighted items, we want to create a generic sample of a certain size that we can later use to estimate the total weight of arbitrary subsets. For this purpose, we propose priority sampling which tested on Internet…
Consider the puzzle: given a number, remove $k$ digits such that the resulting number is as large as possible. Various techniques were employed to derive a linear-time solution to the puzzle: predicate logic was used to justify the…
Let $b \ge 2$ be an integer. Not much is known on the representation in base $b$ of prime numbers or of numbers whose prime factors belong to a given, finite set. Among other results, we establish that any sufficiently large integer which…
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…
What is the number of rolls of fair 6-sided dice until the first time the total sum of all rolls is a prime? We compute the expectation and the variance of this random variable up to an additive error of less than 10^{-4}. This is a…
We investigate the distribution of the digits of quotients of randomly chosen positive integers taken from the interval $[1,T]$, improving the previously known error term for the counting function as $T\to+\infty$. We also resolve some…
We study two types of probability measures on the set of integer partitions of $n$ with at most $m$ parts. The first one chooses the random partition with a chance related to its largest part only. We then obtain the limiting distributions…
In this paper we present a method for producing asymptotic estimates for the number of integers in a given S having only ``small'' prime factors. The conditions that need to be verified are simpler than those required by other methods, and…
The paper describes a prime factorization of the Catalan numbers. Odd prime factors are distributed in layers in accordance with Legendre's formula. The content of each layer is a network of the intervals, Chebyshev's Segments. The primes…
An odd perfect number, N, is shown to have at least nine distinct prime factors. If 3 does not divide N, then N must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect…
We propose a new method, probabilistic divide-and-conquer, for improving the success probability in rejection sampling. For the example of integer partitions, there is an ideal recursive scheme which improves the rejection cost from…
An algorithm for sampling exactly from the normal distribution is given. The algorithm reads some number of uniformly distributed random digits in a given base and generates an initial portion of the representation of a normal deviate in…
In this paper we introduce a simple method of searching for the prime pairs in the famous Goldbach Conjecture. The method, which is based on certain integer identities as well as an observation related to the remainder property, enables us…
We establish asymptotic upper bounds on the number of zeros modulo $p$ of certain polynomials with integer coefficients, with $p$ prime numbers arbitrarily large. The polynomials we consider have degree of size $p$ and are obtained by…
A standard method for finding a rational number from its values modulo a collection of primes is to determine its value modulo the product of the primes via Chinese remaindering, and then use Farey sequences for rational reconstruction.…
We are interested in classifying those sets of primes $\mathcal{P}$ such that when we sieve out the integers up to $x$ by the primes in $\mathcal{P}^c$ we are left with roughly the expected number of unsieved integers. In particular, we…
Generating primes is a fundamental problem in modern cryptography. Deterministic primality tests work well for special integers such as Mersenne or Proth primes, but these forms are quite restrictive. In this paper, we give a direct method…
ABSTRACT. In this article we present a point of view that highlights the importance of finding the upper bounds for prime gaps, in order to solve the twin primes conjecture and the Goldbach conjecture. For this purpose, we present a…