Related papers: Results for Wieferich Primes

On the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers.

The twin prime conjecture asserts that there are infinitely many pairs of primes that differ by two. While recent advances have improved our understanding of bounded prime gaps, the conjecture remains unresolved. This paper refines the…

Let $s$, $\ell$ be two integers such that $2\le s\le \ell-1$, $\ell\ge 3$. We prove that a suitable asymptotic formula for the average number of representations of integers $n=\sum_{i=1}^{s} p_{i}^{\ell}$, where $p_i$, $i=1,\dotsc,s$, are…

In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals (x,x+x^epsilon] is about x^epsilon/log x and the second says that the number of primes…

We obtain an asymptotic formula for the number of primes $p\leq x_1$, $p\leq x_2$ such that $p_1(p_2+a)\equiv l \pmod q$ with $(a,q)=(l,q)=1$, $q\leq x^{\kappa_0}$, $x_1\geq x^{1-\alpha}$, $x_2\geq x^{\alpha}$, $$…

We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length $y$ around $x$, where $y\ll (\log x)^2$. In particular we conjecture that the maximum grows surprisingly slowly as $y$…

The Piatetski-Shapiro sequences are of the form $\mathcal{N}_{c} := (\lfloor n^{c} \rfloor)_{n=1}^\infty$, where $\lfloor \cdot \rfloor$ is the integer part. It is expected that there are infinitely many primes in a Piatetski-Shapiro…

Let $p>3$ be a prime and $b\ge 2$ an integer such that $p$ does not divide $b$. Then $1/p$ has a periodic digit expansion with respect to the basis $b$. The length $q$ of the period is the (multiplicative) order of $b$ mod $p$. In the case…

In this paper, we show a new upper bound of prime gaps, that is the gap between a prime number and its consecutive prime number. We show that the gap between a prime number $p_n$ and its consecutive prime number is not larger than…

Let $x \geq 1$ be a large number, let $f(x) \in \mathbb{Z}[x]$ be a prime polynomial of degree $\text{deg}(f)=m$, and let $u\ne \pm 1, v^2$ be a fixed integer. Assuming the Bateman-Horn conjecture, an asymptotic counting function for the…

For any positive integer r, let pi_{2r}(x) denote the number of prime pairs (p, p+2r) with p not exceeding (large) x. According to the prime-pair conjecture of Hardy and Littlewood, pi_{2r}(x) should be asymptotic to 2C_{2r}li_2(x) with an…

We present in this work a heuristic expression for the density of prime numbers. Our expression leads to results which possesses approximately the same precision of the Riemann's function in the domain that goes from 2 to 1010 at least.…

We establish a lower bound of 2/p(p-1) for the asymptotic density of the Motzkin numbers divisible by a general prime number p > 3. We provide a criteria for when this asymptotic density is actually 1. We also provide a partial…

We prove new results on the additive theory of reversed primes $\overleftarrow{p}$; that is, primes $p$ which are written backwards in a fixed base $b\geq 2$. In particular, we study a variant of Goldbach's conjecture, looking at…

We prove a kind of "almost all symmetry" result for the primes, i.e. we give non-trivial bounds for the "symmetry integral", say $I_{\Lambda}(N,h)$, of the von Mangoldt function $\Lambda(n)$ ($:= \log p$ for prime-powers $n=p^r$, 0…

Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c\:\frac{\log p_n\: \log_2 p_n\: \log_4 p_n}{\log_3 p_n}\:,$$ with $c$ being a…

Let $A$ be a subset of primes up to $x$. If we assume $A$ is well-distributed (in the Siegel-Walfisz sense) in any arithmetic progressions to moduli $q\leqslant(\log x)^c$ for any $c>0$, then the sumset $A+A$ has density 1/2 in the natural…

We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis.

By establishing an improved level of distribution we study almost primes of the form $f(p,n)$ where $f$ is an irreducible binary form over $\mathbb Z$.

Using a sieve-theoretic argument, we show that almost all gaps $(p_n, p_{n+1})$ between consecutive primes $p_n, p_{n+1}$ contain a natural number $m$ whose least prime factor $p(m)$ is at least the length $p_{n+1} - p_n$ of the gap,…