Related papers: Conjectures on Sums of Consecutive Primes

Let $n,k\in\mathbb{N}$ and let $p_{n}$ denote the $n$th prime number. We define $p_{n}^{(k)}$ recursively as $p_{n}^{(1)}:=p_{n}$ and $p_{n}^{(k)}=p_{p_{n}^{(k-1)}}$, that is, $p_{n}^{(k)}$ is the $p_{n}^{(k-1)}$th prime. In this note we…

The famous strongly binary Goldbach's conjecture asserts that every even number $2n \geq 8$ can always be expressible as the sum of two distinct odd prime numbers. We use a new approach to dealing with this conjecture. Specifically, we…

Using evaluations of the difference between consecutive primes we develop another way of estimating of the number of primes in the interval $(n, 2n)$. We also discuss the ultra Cramer conjecture, $p_{n+1} - p_n = O(log^{1+\epsilon}p_n)$…

Let $m$, $r$ and $n$ be positive integers. We denote by ${\bf k}\vdash n$ any tuple of odd positive integers ${\bf k}=(k_1,\dots,k_t)$ such that $k_1+\dots+k_t=n$ and $k_j\ge 3$ for all $j$. In this paper we prove that for every…

Let $m$ be any positive integer and let $\delta_1,\delta_2\in\{1,-1\}$. We show that for some constanst $C_m>0$ there are infinitely many integers $n>1$ with $p_{n+m}-p_n\le C_m$ such that $$\left(\frac{p_{n+i}}{p_{n+j}}\right)=\delta_1\…

It is shown that the first $n$ prime numbers $p_1,...,p_n$ determine the next one by the recursion equation $$ p_{n+1} =\lim\limits_{s\to +\infty} [\prod\limits^n_{k=1} (1-\frac{1}{p^s_k}) \sum\limits^\infty_{j=1} \frac{1}{j^s} -1]^{-1/s}.…

Let $\mathcal{R}_k(n)$ be the number of representations of an integer $n$ as the sum of a prime and a $k$-th power. Define E_k(X) := |\{n \le X, n \in I_k, n\text{not a sum of a prime and a $k$-th power}\}|. Hardy and Littlewood conjectured…

Let $p$ be an odd prime. In the paper we collect the author's various conjectures on congruences modulo $p$ or $p^2$, which are concerned with sums of binomial coefficients, Lucas sequences, power residues and special binary quadratic…

In this paper, we study some supercongruences involving the sequence $$ t_n(x)=\sum_{k=0}^n\binom{n}{k}\binom{x}{k}\binom{x+k}{k}2^k $$ and solve some open problems. For any odd prime $p$ and $p$-adic integer $x$, we determine…

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…

We study a recursively defined sequence which is constructed using the least common multiple. It has been conjectured that every term of that sequence is $1$ or a prime. In this paper we show that this claim is connected to a strong version…

In this article we study in depth the Dirichlet theorem, which states that if a, b are relative prime integers, the sequence p = an + b contains infinite prime numbers, we simplify and generalize this theorem, we enunciate some special…

Mathematicians has been trying to prove the weak Goldbach's conjecture by adding prime numbers, as stated in the conjecture. However, we believe that the solution does not need to be analytically solved. Instead of trying to add prime…

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…

We study an LCM-based analogue of Rowland's GCD-based prime-generating recurrence, introduced by the author in 2008. The multiplicative increments of this sequence are conjectured always to be $1$ or prime, but a complete proof requires a…

Dickson conjectured that a set of polynomials will take on infinitely many simultaneous prime values. Later others, such as Hardy and Littlewood, gave estimates for the number of these primes. In this article we look at this conjecture,…

It is known that there are infinitely-many prime numbers which take the form of a polynomial of degree one with integer coefficients, this is Dirichlet's theorem. We use an elementary sieving argument together with bounds on the prime…

In this paper, using the well known fact that the series of reciprocals of primes diverges, we obtain a general inequality for gaps of consecutive primes that holds for infinitely many primes. As it is shown the key ingredient for this…

It is shown that if every odd integer $n > 5$ is the sum of three primes, then every even integer $n > 2$ is the sum of two primes. A conditional proof of Goldbach's conjecture, based on Cram\'er's conjecture, is presented. Theoretical and…

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}(…