Related papers: Proof of the Goldbach conjecture in a more stringe…

In 1973, J.-R. Chen showed that every large even integer is a sum of a prime and a product of at most two primes. In this paper, the author indicates and fixes the issues in a simplified proof of this result given by Pan et al.

The ternary Goldbach conjecture states that every odd number n>=7 is the sum of three primes. The estimation of sums of the form \sum_{p\leq x} e(\alpha p), \alpha = a/q + O(1/q^2), has been a central part of the main approach to the…

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.

The ternary Goldbach conjecture states that every odd number $n\geq 7$ is the sum of three primes. The estimation of the Fourier series $\sum_{p\leq x} e(\alpha p)$ and related sums has been central to the study of the problem since Hardy…

We show that for any set $D$ of at least two digits in a given base $b$, there exists a $\delta(D,b)>0$ such that within the set $\mathcal{A}$ of numbers whose digits base $b$ are exclusively from $D$, the number of even integers in…

We introduce a collection of polynomials $F_N$, associated to each positive integer $N$, whose divisibility properties yield a reformulation of the Goldbach conjecture. While this reformulation certainly does not lead to a resolution of the…

In the paper, there are new found methods to determine the range of every exceptional element in exceptional set, we can solve Twin primes problem and Goldbach Conjecture problem basically.

By means of $q$-series, we prove that any natural number is a sum of an even square and two triangular numbers, and that each positive integer is a sum of a triangular number plus $x^2+y^2$ for some integers $x$ and $y$ with $x\not\equiv y…

We consider the representation of primes as a sum of a prime and twice a triangular number. We prove that a subset of the primes having density 1 is expressible in this form. We conjecture that every odd prime number is expressible as a sum…

For any pair of real numbers $(i,j)$ with $0<i,j<1$ and $i+j=1$, we prove that the set of $p$-adic mixed $(i,j)$-badly approximable numbers $\bad_p(i, j)$ is 1/2-winning in the sense of Schmidt's game. This improves a recent result of…

We prove versions of Goldbach conjectures for Gaussian primes in arbitrary sectors. Fix an interval $\omega \subset \mathbb{T}$. There is an integer $N_\omega $, so that every odd integer $n$ with $N(n)>N_\omega $ and $\text{dist}(…

A new explicit formula is proved for the contribution of the major arcs in the Goldbach and Generalized Twin Prime Problem, in which the level of the major arcs can be chosen very high. This will have many applications in the approximations…

Let $N$ denote a sufficiently large even integer and $x$ denote a sufficiently large integer, we define $D_{1,2}(N)$ as the number of primes $p$ that such that $N - p$ has at most 2 prime factors. In this paper, we show that $D_{1,2}(N)…

In this paper we present some observations about the well-known Goldbach conjecture. In particular we list and interpret some numerical results which allow us to formulate a relation between prime numbers and even integers. We can also…

Consider the set of all natural numbers that are co-prime to primes less than or equal to a given prime. Then given a consecutive pair of numbers in that set with an arbitrary even gap, we prove there exists an unbounded number of actual…

A specific Goldbach partition of any given even number greater than 6 can be found definitely.

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…

Let $A$ be a subset of primes up to $x$. If we assume $A$ is well-distributed (in the Siegel-Walfisz sense) in any arithmetic progressions to moduli $q\leqslant(\log x)^c$ for any $c>0$, then the sumset $A+A$ has density 1/2 in the natural…

In this article, we introduce congruential Euler numbers, which are a further generalization of generalized Euler numbers. We prove the $p$-adic congruences of congruential Euler numbers, which include answers to a conjecture related to…

Due to the distribution of primes among integers, we establish an upper bound for the probability $\mathbb{P}_n$ that the Goldbach conjecture fails. Assuming the conjecture holds true for all even number less than $2N$, we prove this…