Related papers: What is the smallest prime?

Let $\alpha$ be a real number such that $1< \alpha <2$ and let $x_0=x_0(\alpha)$ be a {\rm(}unique{\rm)} positive solution of the equation $$ x^{\alpha-1} -\frac{\pi}{e^2\sqrt{3}}x +1=0. $$ Then we prove that for each positive integer…

Let $p_n$ be $n$th prime, and let $(S_n)_{n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum_{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding…

It is shown that the first $n$ prime numbers $p_1,...,p_n$ determine the next one by the recursion equation $$ p_{n+1} =\lim\limits_{s\to +\infty} [\prod\limits^n_{k=1} (1-\frac{1}{p^s_k}) \sum\limits^\infty_{j=1} \frac{1}{j^s} -1]^{-1/s}.…

In this paper, we find all integers $x$ such that $x^{2}-1$ has only prime factors smaller than 100. This gives some interesting numerical corollaries. For example, for any positive integer $n$ we can find the largest positive integer $x$…

Bertrand's postulate establishes that for all positive integers $n>1$ there exists a prime number between $n$ and $2n$. We consider a generalization of this theorem as: for integers $n\geq k\geq 2$ is there a prime number between $kn$ and…

For n=1,2,3,... define S(n) as the smallest integer m>1 such that those 2k(k-1) mod m for k=1,...,n are pairwise distinct; we show that S(n) is the least prime greater than 2n-2 and hence the value set of the function S(n) is exactly the…

Let $ \lfloor {x} \rfloor $ denote the greatest integer less than or equal to a real number $x$. Given real numbers $0<\alpha_1 < \alpha_2 < \cdots< \alpha_k < 1$ satisfying a certain condition, we show that there are infinitely many…

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…

Prime number multiplet classifications and patterns are extended to negative integers. The extension from prime numbers to single prime powers is also studied. Prime number septets at equal distance are given. It is also shown that each…

The twin primes conjecture is a very old problem. Tacitly it is supposed that the primes it deals with are finite. In the present paper we consider three problems that are not related to finite primes but deal with infinite integers. The…

I give some claims on primorial prime numbers for interested readers in number theory.

If $a>b$ and $n>1$ are positive integers and $a$ and $b$ are relatively prime integers, then a large Zsigmondy prime for $(a,b,n)$ is a prime $p$ such that $p \,|\, a^n-b^n$ but $p \,\nmid \, a^m-b^m$ for $1 \leq m < n$ and either $p^2 \, |…

A positive integer $n$ is defined to be cyclic if and only if every group of size $n$ is cyclic. Equivalently, $n$ is cyclic if and only if $n$ is relatively prime to the number of positive integers less than $n$ that are relatively prime…

In this work we show that the prime distribution is deterministic. Indeed the set of prime numbers P can be expressed in terms of two subsets of N using three specific selection rules, acting on two sets of prime candidates. The prime…

This work proposes elementary proofs of several related primes counting problems, based on an elementary weighted sieve. The subsets of primes considered here are the followings: the subset of twin primes PT = {p and p + 2 are primes}, the…

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x \ge R_n$, then there are at least $n$ primes in the interval $(x/2,x]$. For example, Bertrand's postulate is $R_1 = 2$. Ramanujan proved that $R_n$ exists and…

Logarithmic gaps have been used in order to find a periodic component of the sequence of prime numbers, hidden by a random noise (stochastic or chaotic). The recovered period for the sequence of the first 10000 prime numbers is equal to…

Under Cram\'er's conjecture concerning the prime numbers, we prove that for any $x>1$, there exists a real $A=A(x)>1$ for which the formula $[A^{n^x}]$ (where $[]$ denotes the integer part) gives a prime number for any positive integer $n$.…

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…

When are all positions of a game numbers? We show that two properties are necessary and sufficient. These properties are consequences of that, in a number, it is not an advantage to be the first player. One of these properties implies the…