Related papers: Primes in intervals of bounded length
In this paper we prove that the binary Goldbach conjecture for sufficiently large even integers would follow under the assumption that both the Elliott-Halberstam conjecture and a variant of the Elliott-Halberstam conjecture twisted by the…
By creating a new method, the author proved the well-known world's baffling problems Goldbach conjecture, twin primes conjecture, the Proposition (C) and the Proposition $n^2+1$.
It is the purpose of this thesis to enunciate and prove a collection of explicit results in the theory of prime numbers. First, the problem of primes in short intervals is considered. We prove that there is a prime between consecutive cubes…
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 show the primes have level of distribution $66/107\approx 0.617$ using triply well-factorable weights. This gives the highest level of distribution for primes in any setting, improving on the prior record level $3/5=0.60$ of Maynard. We…
The author gives nontrivial upper and lower bounds for the number of primes in the interval $[x - x^{\theta}, x]$ for some $0.52 \leqslant \theta \leqslant 0.525$, showing that the interval $[x - x^{0.52}, x]$ contains prime numbers for all…
We improve Bombieri's asymptotic sieve to localise the variables. As a consequence, we prove, under a Elliott-Halberstam conjecture, that there exists an infinity of twins almost prime. Those are prime numbers $p$ such that for all…
In 1737 Leonard Euler gave what we often now think of as a new proof, based on infinite series, of Euclid's theorem that there are infinitely many prime numbers. Our short paper uses a simple modification of Euler's argument to obtain new…
We bring to bear an empirical model of the distribution of twin primes and produce two distinct results. The first is that we can make a quantitative probabilistic prediction of the occurrence of gaps in the sequence of twins within the…
This note discusses the existence of prime numbers in short intervals. An unconditional elementary argument seems to prove the existence of primes in the short intervals [x, x + y], where y >= x^(1/2)(log x)^e, e > 0, and a sufficiently…
Some new results concerning the equation $\sigma(N)=aM, \sigma(M)=bN$ are proved. As a corollary, there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors.
Let $\mathcal{P}$ be the set of all primes and $\pi(x)$ be the number of primes up to $x$. For any $n\ge 2$, let $P^+(n)$ be the largest prime factor of $n$. For $0<c<1$, let $$T_c(x)=\#\{p\le x:p\in \mathcal{P},P^+(p-1)\ge p^c\}.$$ In this…
From known effective bounds on the prime counting function of the form \[ |\pi(x)-\mathrm{Li}(x)| < a \;x \;(\ln x)^{b} \; \exp\left(-{c}\; \sqrt{\ln x}\right); \qquad (x \geq x_0); \] it is possible to establish exponentially tight…
We prove an explicit analogue of Legendre's conjecture for almost primes. Namely, for every integer $n \geq 1$, the interval $(n^2,(n+1)^2)$ contains an integer having at most $3$ prime factors, counted with multiplicity. This improves the…
We use a homogeneous catalog of 42,000 main-sequence wide binaries identified by Gaia to measure the mass ratio distribution, p(q), of binaries with primary masses $0.1<M_1/M_{\odot}<2.5$, mass ratios $0.1 \lesssim q<1$, and separations…
The world of primes has many gaps between evidence and theorems. Here, we review Legendre's conjecture on primes between consecutive squares and recent progress on the weaker question of primes between consecutive larger powers. Assuming…
n 1937 Ivan Vinogradov proved the three prime sum version of the Goldbach Conjecture, often called the weak form of Goldbach Conjecture. And that it holds for "sufficiently large" odd natural numbers. In this work we use Dirichlet Theorem,…
We adopt an empirical approach to the characterization of the distribution of twin primes within the set of primes, rather than in the set of all natural numbers. The occurrences of twin primes in any finite sequence of primes are like…
In this short paper we present an elementary proof of the infinitude of primes. Our proof is similar in spirit to Euler's proof that the reciprocals of primes diverges and only uses tools from elementary number theory and calculus. In…
We propose the formula for the number of pairs of consecutive primes $p_n, p_{n+1}<x$ separated by gap $d=p_{n+1}-p_n$ expressed directly by the number of all primes $<x$, i.e. by $\pi(x)$. As the application of this formula we formulate 7…