Related papers: On the gap between primes

We present effective upper bounds on the symmetric bilinear complexity of multiplication in extensions of a base finite field Fp2 of prime square order, obtained by combining estimates on gaps between prime numbers together with an optimal…

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…

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, $p_{n+1} - p_n = O(log^{1+\epsilon}p_n)$…

We derive explicit formulas for integrals of certain symmetric polynomials used in Keiju Sono's multidimensional sieve of $E_2$-numbers, i.e., integers which are products of two distinct primes. We use these computations to produce the…

Assuming that there exist (infinitely many) Siegel zeros, we show that the (Rosser-)Jurkat-Richert bounds in the linear sieve cannot be improved, and similarly look at Iwaniec's lower bound on Jacobsthal's problem, as well as minor…

For $x>0$ let $\pi(x)$ denote the number of primes not exceeding $x$. For integers $a$ and $m>0$, we determine when there is an integer $n>1$ with $\pi(n)=(n+a)/m$. In particular, we show that for any integers $m>2$ and $a\le\lceil…

We develop machinery to explicitly determine, in many instances, when the difference $x^2-y^n$ is divisible only by powers of a given fixed prime. This combines a wide variety of techniques from Diophantine approximation (bounds for linear…

A positive integer is called an $E_j$-number if it is the product of $j$ distinct primes. We prove that there are infinitely many triples of $E_2$-numbers within a gap size of $32$ and infinitely many triples of $E_3$-numbers within a gap…

In this paper, we consider the simultaneous representation of pairs of sufficiently large integers. We prove that every pair of large positive odd integers can be represented in the form of a pair of one prime, four cubes of primes and 231…

In this paper we review the properties of families of numbers of the form $6n\pm1$, with $n$ integer (in which there are all prime numbers greater than 3 and other compound numbers with particular properties) to later use them in a new…

We show that a positive proportion of all gaps between consecutive primes are small gaps. We provide several quantitative results, some unconditional and some conditional, in this flavour.

Let $a_0\in\{0,\dots,9\}$. We show there are infinitely many prime numbers which do not have the digit $a_0$ in their decimal expansion. The proof is an application of the Hardy-Littlewood circle method to a binary problem, and rests on…

We give an estimation of the existence density for the $2d$ different primes by using a new and simple algorithm for getting the $2d$ different primes. The algorithm is a kind of the sieve method, but the remainders are the central numbers…

We investigate the group of points of the $3$-sphere modulo a prime, point out connections to other known groups and the Chebyshev polynomials, and show that there is an infinite series which converges if and only if there are finitely many…

The author proves that there are infinitely many primes $p$ such that $\| \alpha p - \beta \| < p^{-\frac{28}{87}}$, where $\alpha$ is an irrational number and $\beta$ is a real number. This sharpens a result of Jia (2000) and provides a…

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 consider the exceptional set in the binary Goldbach problem for sums of two almost twin primes. Our main result is a power-saving bound for the exceptional set in the problem of representing $m=p_1+p_2$ where $p_1+2$ has at most $2$…

Fix a non-CM elliptic curve $E/\mathbb{Q}$, and let $a_E(p) = p + 1 - \#E(\mathbb{F}_p)$ denote the trace of Frobenius at $p$. The Sato-Tate conjecture gives the limiting distribution $\mu_{ST}$ of $a_E(p)/(2\sqrt{p})$ within $[-1, 1]$. We…

While the twin prime conjecture is still famously open, it holds true in the setting of finite fields: There are infinitely many pairs of monic irreducible polynomials over $\mathbb{F}_q$ that differ by a fixed constant, for each $q \geq…

1. There is no existing any quadratic interval $\eta_{n}:=(n^{2},(n+1)^{2}],$ which contains less than 2 prime numbers. The number of prime numbers within $\eta_{n}$ goes averagely linear with n to infinity. 2. The exact law of the number…