Related papers: Minor arcs for Goldbach's problem

We prove an inverse ternary Goldbach-type result. Let $N$ be sufficiently large and $c>0$ be sufficiently small. If $A_1,A_2,A_3\subset [N]$ are subsets with $|A_1|,|A_2|,|A_3|\geq N^{1/3-c}$, then $A_1+A_2+A_3$ contains a composite number.…

Let $\mathcal{P}$ denote the set of all primes. $P_{1},P_{2},P_{3}$ are three subsets of $\mathcal{P}$. Let $\underline{\delta}(P_{i})$ $(i=1,2,3)$ denote the lower density of $P_{i}$ in $\mathcal{P}$, respectively. It is proved that if…

In this Note, we try to study the relations between the Goldbach Conjecture and the least prime number in an arithmetic progression. We give a new weakened form of the Goldbach Conjecture. We prove that this weakened form and a weakened…

We formulate some refinements of Goldbach's conjectures based on heuristic arguments and numerical data. For instance, any even number greater than 4 is conjectured to be a sum of two primes with one prime being 3 mod 4. In general, for…

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…

We prove that there exists a k_0>0 such that every sufficiently large odd integer n with 3\mid n can be represented as p_1+p_2+p_3, where p_1,p_2 are Chen's primes and p_3 is a prime with p_3+2 has at most k_0 prime factors.

The binary Goldbach conjecture asserts that every even integer greater than $4$ is the sum of two primes. In a preceding paper we have proved that there exists a positive integer $K_\alpha$ such that every even integer $x > p_k^2$ can be…

The Goldbach conjecture that, every even integer is the sum of two primes, has been open since 1742. This paper details a road map to a proof of Goldbachs conjecture based on a function that estimates the number of Goldbach pairs. It is…

In this work we use the number classification in families of the form 6n+1, and 6n+5 with n integer (Such families contain all odd prime numbers greater than 3 and other compound numbers related with primes). We will use this kind of…

Let $$\gamma^*=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx 0.919\ldots)\:.$$ Let $\gamma^*<\gamma_0\leq 1$, $c_0=1/\gamma_0$ be fixed. Let also $a_0\in\{0,1,\ldots, 9\}$.\\ We prove on assumption of the Generalized Riemann…

For A,epsilon>0 and any sufficiently large odd n we show that for almost all k up to n^{1/5-epsilon} there exists a representation n=p1+p2+p3 with primes in residue classes b1,b2,b3 mod k for almost all admissible triplets b1,b2,b3 of…

We answer the question positively. In fact, we believe to have proved that every even integer $2N\geq3\times10^{6}$ is the sum of two odd distinct primes. Numerical calculations extend this result for $2N$ in the range $8-3\times10^{6}$.…

Let $p_{1}$, ..., $p_{k}$ be the first $k$ odd primes in succession. Let $n$ be an even integer such that $n > p_{k}$. We conjecture that if none of $n - p_{1}$, ..., $n - p_{k}$ are prime, then at least one of them has a prime factor which…

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…

ABSTRACT. In this article we present a point of view that highlights the importance of finding the upper bounds for prime gaps, in order to solve the twin primes conjecture and the Goldbach conjecture. For this purpose, we present a…

For two odd primes $p$ and $q$ such that $p<q$, let $A(p,q):=(a_k)_{k=1}^{\infty}$ be the arithmetic progression whose $k$th term is given by $a_k=(k-1)(q-p)+p$ (i.e., with $a_1=p$ and $a_2=q$). Here we conjecture that for every positive…

Let $\eta$ be a quadratic irrationality. The variant of a ternary problem of Goldbach involving primes such that $a<\{\eta p\}<b$, where $a$ and $b$ are arbitrary numbers of the interval $(0,1)$, solved in this paper.

Let $$\gamma^*:=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx 0.919\ldots)\:,\ \gamma^*<\frac{1}{c_0}\leq 1\:.$$ Let $\gamma^*<\gamma_0\leq 1$, $c_0=1/\gamma_0$ be fixed. Let also $a_0\in\{0,1,\ldots, 9\}$. In [23] we proved…

The idea of generating prime numbers through sequence of sets of co-primes was the starting point of this paper that ends up by proving two conjectures, the existence of infinitely many twin primes and the Goldbach conjecture. The main idea…

In this paper, a simple explanation for the Goldbach Conjecture is given. We have shown that the probability of violating the conjecture not only for the prime numbers, but also for any subset of natural numbers whose distribution is…