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For integers x and k, let T(x;2k) denote the number of twin prime pairs (p,p+2k) with a distance 2k<=2x**0.5 and p<=x (not p+2k<=x). Let Tg(x;2x**0.5) denote the average of T(x;2k) for all 2k<=2x**0.5. Logically, T(x;2k) should be a…
There is extensive numerical support for the prime-pair conjecture (PPC) of Hardy and Littlewood (1923) on the asymptotic behavior of pi_{2r}(x), the number of prime pairs (p,p+2r) with p not exceeding x. However, it is still not known…
A pair of odd primes is said to be symmetric if each prime is congruent to one modulo their difference. A theorem from 1996 by Fletcher, Lindgren, and the third author provides an upper bound on the number of primes up to x that belong to a…
We present a novel approach to the Twin Prime Conjecture, basing on the $6x \pm 1$ representation of primes. By defining so-called twin prime generators $x \in \N$, for which both $6x - 1$ and $6x + 1$ are prime, we reformulate the…
We introduce Wilson's theorem and Clement's result and present a necessary and sufficient condition for p and p+2k to be primes where k is a positive integer. By using Simiov's Theorem, we derive an improved version of Clement's result and…
Twin prime number problem is mainly the structure of the twin prime numbers and whether there are infinitely many prime twins group. In this paper, by constructing a special cluster number set(see formula(2.3)in the paper), proves that the…
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…
Goldbach`s Conjecture, "every even number greater than 2 can be expressed as the sum of two primes" is renamed Goldbach`s Rule for it can not be otherwise. The conjecture is proven by showing that the existence of prime pairs adding to any…
In 1845, Bertrand conjectured that twice any prime strictly exceeds the next prime. Tchebichef proved Bertrand's postulate in 1850. In 1934, Ishikawa proved a stronger result: the sum of any two consecutive primes strictly exceeds the next…
By creating a new method, the author proved the well-known world's baffling problems Goldbach conjecture, twin primes conjecture, the Proposition (C) and the Proposition $n^2+1$.
The famous strongly binary Goldbach's conjecture asserts that every even number $2n \geq 8$ can always be expressible as the sum of two distinct odd prime numbers. We use a new approach to dealing with this conjecture. Specifically, we…
We establish an asymptotic formula for the number of k-difference twin primes associated with the Hawkins random sieve, which is a probabilistic model of the Eratosthenes sieve. The formula for k=1 was obtained by Wunderlich [Acta Arith. 26…
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…
This document seeks to prove there are infinitely many primes whose difference is 2, referred to as twin prime pairs. This proof's methodology involves constructing a function that approximates the number of positive integers, less than a…
The twin prime conjecture asserts that there are infinitely many pairs of primes that differ by two. While recent advances have improved our understanding of bounded prime gaps, the conjecture remains unresolved. This paper refines the…
In this paper, we make some conjectures on prime numbers that are sharper than those found in the current literature. First we describe our studies on Legendre's Conjecture which is still unsolved. Next, we show that Brocard's Conjecture…
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
We present some new ideas on important problems related to primes. The topics of our discussion are: simple formulae for primes, twin primes, Sophie Germain primes, prime tuples less than or equal to a predefined number, and their…
We address the question of the infinitude of twin and cousin prime pairs from a probabilistic perspective. Our approach partitions the set of integer numbers greater than $2$ in finite intervals of the form $[p_{n-1}^2,p_n^2)$, $p_{n-1}$…
This paper analyzes the emergence and distribution of potential twin primes, pairs of integers that are both relatively prime to the first n primes or to a given set M of primes, and which are the breeding grounds of true twin primes. It…