Related papers: Dealing with prime numbers I.: On the Goldbach con…
We study additive properties of consecutive prime numbers and the primality of the sums they generate. For a given prime number $p_n$, we consider the sums \[ S_k(p_n) = p_n + p_{n+1} + \cdots + p_{n+k-1}, \] where $k \ge 3$ is an odd…
We give a more comrehensive treatment of Chen's double sieve and improve related constants in Goldbach's conjecture and the twin prime problem.
$\Theta$ function is defined based upon Kronecher symbol. In light of the principle of inclusion-exclusion, $\Theta$ function of sine function is used to denote the distribution of composites and primes. The structure of Goldbach Conjecture…
Six conjectures on pairs of consecutive primes are listed in this paper, together with examples for each case.
In this paper we study a sequence involving the prime numbers by deriving two asymptotic formulas and finding new upper and lower bounds, which improve the currently known estimates.
In this paper we study the integrals of fractional parts of given functions, and develop some new tools to understand the behaviour of prime differences. We demonstrate how simply some seemingly difficult conjectures related to prime…
In this article, we investigate how Euler might have been led to conjecture the Prime Number Theorem, based on what he knew. We also speculate on why he did not do so.
We survey the classical results on the prime number theorem
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…
Legendre's conjecture states that there is a prime number between n^2 and (n+1)^2 for every positive integer n. We consider the following question : for all integer n>1 and a fixed integer k<=n does there exist a prime number such that kn <…
This paper investigates the use of the number of Goldbach triples, or the number of three prime partitions of an odd number, for use in the generation and distribution of cryptographic keys. In addition to presenting randomness properties…
On the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers.
A well-known conjecture asserts that, for any given positive real number $\lambda$ and nonnegative integer $m$, the proportion of positive integers $n \le x$ for which the interval $(n,n + \lambda\log n]$ contains exactly $m$ primes is…
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…
The goal of this paper is to describe an elementary combinatorial heuristic that predicts Hardy and Littlewood's extended Goldbach's conjecture. We examine common features of other heuristics in additive prime number theory, such as…
In this article we present method of solving some additive problems with primes. The method may be employed to the Goldbach-Euler conjecture and the twin primes conjecture. The presented method also makes it possible to obtain some…
We prove some theorems which give sufficient conditions for the existence of prime numbers among the terms of a sequence which has pairwise relatively prime terms.
We create a simple test for distinguishing between sets of primes and random numbers using just the sum-of-digits function. We find that the sum-of-the-digits of prime numbers does not have an equal probability of being odd or even. The…
In this paper we derive congruences expressing Bell numbers and derangement numbers in terms of each other modulo any prime.
In this paper, we consider pairs of a prime and a prime power with a fixed difference. We prove an average result on the distribution of such pairs. This is a partial improvement of the result of Bauer (1998).