Related papers: Theorems on twin primes-dual case
We investigate the group of points of the $3$-sphere modulo a prime, point out connections to other known groups and the Chebyshev polynomials, and show that there is an infinite series which converges if and only if there are finitely many…
This talk is divided into two parts. The first part reviews some of the duality relationships between superstring theories. These relationships are interpreted as providing evidence for the existence of a unique underlying fundamental…
In this paper proof of the twin prime conjecture is going to be presented. Originally very difficult problem (in observational space) has been transformed into a simpler one (in generative space) that can be solved. It will be shown that…
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 introduce two primality tests based on new divisibility properties of binomial coefficients. These new properties were enunciated and proved in previous work. We also study two similar tests that can be obtained from…
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,…
Extending the works of Alladi and Sweeting and Woo, we state and prove the general higher order duality between prime ideals in number rings. We then use the second order duality to obtain the a new formula for the Chebotarev Density…
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 $n^2+1$.
We prove that there are infinitely often pairs of primes much closer than the average spacing between primes - almost within the square root of the average spacing. We actually prove a more general result concerning the set of values taken…
In this paper, we derive a new proof on some sharp double integral inequalities of the Hermite-Hadamard type. Our approach is mainly based on well-known Taylor's theorem with the integral remainder.
I review the proposal of Berenstein-Douglas for a completely general definition of Seiberg duality. To give evidence for their conjecture I present the first example of a physical dual pair and explicitly check that it satisfies the…
The theorem presented in this paper allows the creation of large prime numbers (of order up to o(n^2)) given a table of all primes up to n.
We provide two new proofs of the infinitude of prime numbers, using the additive Ramsey-theoretic result known as Folkman's theorem (alternatively, one can think of these proofs as using Hindman's theorem). This adds to the existing…
In the paper, the occurrence of zeros and ones in the binary expansion of the primes is studied. In particular the statement in the title is established. The proof is unconditional.
We present a new topological proof of the infinitude of prime numbers with a new topology. Furthermore, in this topology, we characterize the infinitude of any non-empty subset of prime numbers.
Several examples of generalized number systems are constructed to compare various conditions occurring in the literature for the prime number theorem in the context of Beurling generalized primes.
In this paper, we prove the conjecture that if there is an odd perfect number, then there are infinitely many of them.
This article is a collected information from some books and papers, and in most cases the original sentences is reserved about twin prime conjecture.
We first give a bijective proof of Gould's identity in the model of binary words. Then we deduce Rothe's identity from Gould's identity again by a bijection, which also leads to a double-sum extension of the $q$-Chu-Vandermonde formula.
Starting from a generalization of a recent result on self-duality we systematically analyze self-dual models. We find a criterion to judge whether a given model is self-dual or not. With this tool we construct some new self-dual pairs,…