Related papers: The ternary Goldbach problem
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…
In this paper we propose an alternative formulation of the binary and ternary Goldbach conjectures as the systems of equations involving the Euler $\phi$-function.
Prime numbers have attracted the attention of mathematiciansand enthusiasts for millenniums due to their simple definition and remarkable properties. In this paper, we study primorial numbers (the product of the first prime numbers) to…
Using the fact that the number of combinations $p_{1}$, $p_{2}$, where $p_{1}$ and $p_{2}$ are odd primes, with $p_{1} \leq p_{2}$ and $p_{1} + p_{2} \leq 2N$ is equal to the total number of Goldbach pairs for all the even integers from 6…
Goldbach's comet is the plot of the Goldbach function g(n), in the interval [3, N], with a large positive integer N. The function g(n) counts the number of different ways in which 2n can be expressed as the sum of two odd primes. An…
We establish Bombieri-Vinogradov's type result for the number of solutions of the ternary Goldbach problem with primes from arithmetic progressions.
We compute all primes up to $6.25\times 10^{28}$ of the form $m^2+1$. Calculations using this list verify, up to our bound, a less famous conjecture of Goldbach. We introduce `Goldbach champions' as part of the verification process and…
Following an idea of Rowland we give a conjectural way to generate increasing sequences of primes using algorithms involving the gcd. These algorithms seem not so useless for searching primes since it appears we found sometime primes much…
We prove that if A is a subset of the primes, and the lower density of A in the primes is larger than 5/8, then all sufficiently large odd positive integers can be written as the sum of three primes in A. The constant 5/8 in this statement…
A celebrated conjecture of Hardy and Littlewood provides with an asymptotic formula for the counting function of the twin primes. We give an unconditional proof of such a formula by means of a finite Ramanujan expansion of the counting…
We examine the problem of writing every sufficiently large even number as the sum of two primes and at most $K$ powers of 2. We outline an approach that only just falls short of improving the current bounds on $K$. Finally, we improve the…
Improving earlier estimates of several authors we show that the number E(X) of Goldbach exceptional even integers (that is, even integers which cannot be written as the sum of two primesw) below X satisfies tho bound E(X) < X^0.72 for…
In this paper, we develop the method of circle of partitions and associated statistics. As an application we prove conditionally the binary Goldbach conjecture. We develop a series of steps to prove the binary Goldbach conjecture in full.…
Let $\mathcal{P}=\{p_1,p_2,...\}$ be the set of all odd primes arranged in increasing order. A Goldbach partition of the even integer $2k>4$ is a way of writing it as a sum of two primes from $\mathcal{P}$ without regard to order. Let…
In this paper we introduce a simple method of searching for the prime pairs in the famous Goldbach Conjecture. The method, which is based on certain integer identities as well as an observation related to the remainder property, enables us…
In this note, we try to understand the recent development on the Waring-Goldbach problem involving cubes of primes. Especially, we want to determine whether integers that are either primes, squares of primes, cubes of primes, or a cube of…
Ordinary binary multiplication of natural numbers can be generalized in a non-trivial way to a ternary operation by considering discrete volumes of lattice hexagons. With this operation, a natural notion of `3-primality' -- primality with…
In this paper, we prove that every pair of sufficiently large odd integers can be represented in the form of a pair of one prime, four prime cubes and $48$ powers of $2$.
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 survey based on our earlier paper ("The 'Vertical' Generalization of the Binary Goldbach's Conjecture as Applied on 'Iterative' Primes with (Recursive) Prime Indexes (i-primeths)" [11]), a paper in which we have proposed a…