Related papers: Large gaps between consecutive prime numbers conta…

Combining the mollifiers, we exhibit other choices of coefficients that improve the results on large gaps between the zeros of the Riemann zeta-function. Precisely, assuming the Generalized Riemann Hypothesis (GRH), we show that there exist…

In a recent work Friedlander studied the problem of how large consecutive prime gaps should be in order that the sum of the reciprocals should be divergent. Supposing a very deep Hypothesis, a generalization of the Hardy--Littlewood prime…

We establish the existence of infinitely many \emph{polynomial} progressions in the primes; more precisely, given any integer-valued polynomials $P_1, >..., P_k \in \Z[\m]$ in one unknown $\m$ with $P_1(0) = ... = P_k(0) = 0$ and any $\eps…

We determine the complete list of the gaps between successive elements of the multiplication table of the first N integers.

We show that for every positive integer $k$, there exist $k$ consecutive primes having the property that if any digit of any one of the primes, including any of the infinitely many leading zero digits, is changed, then that prime becomes…

We posit that $d_n^2 < 2p_{n+1}$ holds for all $n\geq 1$, where $p_n$ represents the $n$th prime and $d_n$ stands for the $n$th prime gap i.e. $d_n := p_{n+1} - p_n$. Then, the presence of a prime between successive perfect squares, as well…

We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that a positive proportion of consecutive primes are within a…

For every prime $p$, we construct an infinite countable group that contains precisely $p-1$ elements which are not $p$th powers.

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…

We study the first occurrences of gaps between primes in the arithmetic progression (P): $r$, $r+q$, $r+2q$, $r+3q,\ldots,$ where $q$ and $r$ are coprime integers, $q>r\ge1$. The growth trend and distribution of the first-occurrence gap…

Using Duke's large sieve inequality for Hecke Gr{\"o}ssencharaktere and the new sieve methods of Maynard and Tao, we prove a general result on gaps between primes in the context of multidimensional Hecke equidistribution. As an application,…

Let $P(x) \in \mathbb{Z}[x]$ be a polynomial. We give an easy and new proof of the fact that the set of primes $p$ such that $p \mid P(n)$, for some $n \in \mathbb{Z}$, is infinite. We also get analog of this result for some special…

In the present work we prove a common generalization of Maynard-Tao's recent result about consecutive bounded gaps between primes and on the Erd\H{o}s-Rankin bound about large gaps between consecutive primes. The work answers in a strong…

We showed that the prime gap for a prime number p is less than or equal to the prime count of the prime number.

We show that the existence of arithmetic progressions with few primes, with a quantitative bound on "few", implies the existence of larger gaps between primes less than x than is currently known unconditionally. In particular, we derive…

We establish new upper bounds for the length of runs of consecutive positive integers each with exactly $k$ divisors, where $k$ is a given positive integer of some special forms. Also we have found exact values of the maximum possible runs…

Let $p_n$ be $n$th prime, and let $(S_n)_{n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum_{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding…

Let $1<c<d$ be two relatively prime integers, $g_{c,d}=cd-c-d$ and $\mathbb{P}$ is the set of primes. For any given integer $k \geq 1$, we prove that $$\#\left\{p^k\le g_{c,d}:p\in \mathbb{P}, ~p^k=cx+dy,~x,y\in \mathbb{Z}_{\geqslant0}…

Let $p$ be a prime and ${\mathcal{P}_{p}}$ the set of positive integers which are prime to $p$. We establish the following interesting congruence \[\sum\limits_{\begin{smallmatrix} i+j+k={{p}^{r}} i,j,k\in {\mathcal{P}_{p}}…

In this paper we study the problem of detecting prime numbers between all consecutive cubes. Firstly, we use a large computation to show that there is always a prime between $n^3$ and $(n+1)^3$ for $n^3\leq 1.649\cdot 10^{40}$. In addition,…