相关论文: Simple Divisibility Rules for the 1st 1000 Prime N…
In order to avoid unnecessary applications of Miller-Rabin algorithm to the number in question, we resort to trial division by a few initial prime numbers, since such a division take less time. How far we should go with such a division is…
The distribution of prime numbers is here considered. We show a formula for $li^{-1}$ and we study the $\pi(x)$ function and Riemann's hypothesis.
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…
Generalising a previous result, we determine all non-abelian finite simple groups whose order has largest prime divisor not exceeding $10^4$. The computer code for this and similar calculations is made available.
Formula for the nth prime using elementary arithmetical functions based in a previous formula changing the characteristic function of prime numbers.
Let $\mathcal{A}'$ be the set of integers missing any three fixed digits from their decimal expansion. We produce primes in a thin sequence by proving an asymptotic formula for counting primes of the form $p = m^2 + \ell^2$, with $\ell \in…
With the formula of Gandhi you can determine the on $p_n$ immedately subsequent prime $p_{n+1}$ from the knowledge of the primes $p_1, p_2, ... , p_n$. An elementary proof of its trueness will be detailed shown in this paper. Finally the…
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.
To determine whether a number is congruent or not is an old and difficult topic and progress is slow. The paper presents a new theorem when a prime number is a congruent number or not. The proof is not necessarily any simpler or shorter…
A hypothesis is put forward regarding the function $\pi_2(x)$ which describes the distribution of twin primes in the set of natural numbers. The function $\pi_2(x)$ is tested by evaluation and an empirical $\pi_2^{\ast}(x)$ is arrived at,…
In this note, we propose simple summations for primes, which involve two finite nested sums and Bernoulli numbers. The summations can also be expressed in terms of Bernoulli polynomials.
The chronicle of prime numbers travel back thousands of years in human history. Not only the traits of prime numbers have surprised people, but also all those endeavors made for ages to find a pattern in the appearance of prime numbers has…
We consider the regular parts for basic functions of prime numbers with Riemann approximation accuracy.
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 $\lfloor log_{10}(1+n!\sum n)\rceil+1$.
An s-tuple of positive integers are k-wise relatively prime if any k of them are relatively prime. Exact formula is obtained for the probability that s positive integers are k-wise relatively prime.
We have presented a multivariate polynomial function termed as factor elimination function,by which, we can generate prime numbers. This function's mapping behavior can explain the irregularities in the occurrence of prime numbers on the…
10 is the smallest positive integer which is whether solitary or friendly is still an open question in mathematics. In this paper, we provide upper bounds for each of the prime divisors of a friend of 10. This paper is precisely a…
We report on the results of a computer search for primes $p$ which divide an Harmonic number $H_{\lfloor p/N \rfloor}$ with small $N > 1$.
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 the number $373$ all subwords ($3$, $7$, $37$, $73$, and $373$) are prime. Similarly, in $9719$ all subwords are divisible by at most one prime. And similarly again in $7319797913$ all subwords are divisible by at most two primes. These…