Related papers: An upper bound for the prime gap

In this paper we establish a new explicit upper and lower bound for the $n$-th prime number, which improve the currently best estimates given by Dusart in 2010. As the main tool we use some recently obtained explicit estimates for the prime…

This paper discusses prime numbers that are (resp. are not) congruent numbers. Particularly the only case not fully covered by earlier results, namely primes of the form $p=8k+1$, receives attention.

In this note we present a method to bound gaps between primes via the divergence of the series of reciprocals of the prime numbers, a consequence of a version of the Bertrand's test for convergence of series of positive numbers and a…

We study the gaps between consecutive prime numbers directly through Eratosthenes sieve. Using elementary methods, we identify a recursive relation for these gaps and for specific sequences of consecutive gaps, known as constellations.…

It is an open problem whether $ \binom{2n}{n} $ is divisible by 4 or 9 for all $n>256$. In connection with this, we prove that for a fixed uneven $m$ the asymptotic density of $k$'s such that $ m \nmid \binom{2^{k+1}}{2^{k}} $ is 0. To do…

Let $p>1$ be a large prime number and let $x=O((\log p)^2(\log\log p)^5$ be a real number. It is proved that the least consecutive pair of primitive roots $u\ne\pm1, v^2$ and $u+1$ satisfies the upper bound $u\ll x$ in the prime field…

Kemnitz Conjecture [9] states that if we take a sequence of elements in $Z_{p}^{2}$ of length $4p-3$, $p$ is a prime number, then it has a subsequence of length $p$, whose sum is $0$ modulo $p$. It is known that in $Z_{p}^{3}$ to get a…

Let $\{p_j(n)\}_{j=1}^{\omega(n)}$ denote the increasing sequence of distinct prime factors of an integer $n$. We provide details for the proof of a statement of Erd\H{o}s implying that, for any function $\xi(n)$ tending to infinity with…

Let $\delta(p)$ tend to zero arbitrarily slowly as $p\to\infty$. We exhibit an explicit set $\mathcal{S}$ of primes $p$, defined in terms of simple functions of the prime factors of $p-1$, for which the least primitive root of $p$ is $\le…

In this work we prove that the set of the difference of primes is a $\Delta_r^*$-set. The work is based on the recent dramatic new developments in the study of bounded gaps between primes, reached by Zhang, Maynard and Tao.

Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form $[x-x^{0.525},x]$ for large $x$. In this paper, we extend a result of Maynard and Tao concerning small gaps between primes to…

Let $p$ and $q$ be two distinct fixed prime numbers and $(n_i)_{i\geq 0}$ the sequence of consecutive integers of the form $p^a\cdot q^b$ with $a,b\ge 0$. Tijdeman gave a lower bound (1973) and an upper bound (1974) for the gap size…

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…

Let $M(x)$ be the length of the largest subinterval of $[1,x]$ which does not contain any sums of two squareful numbers. We prove a lower bound \[ M(x)\gg \frac{\ln x}{(\ln\ln x)^2} \] for all $x\geq 3$. The proof relies on properties of…

Let s_q(n) denote the base q sum of digits function, which for n<x, is centered around (q-1)/2 log_q x. In Drmota, Mauduit and Rivat's 2009 paper, they look at sum of digits of prime numbers, and provide asymptotics for the size of the set…

We consider polynomial equations, or systems of polynomial equations, with integer coefficients, modulo prime numbers $p$. We offer an elementary approach based on a counting method. The outcome is a weak form of the Lang-Weil lower bound…

We provide a lower bound on the probability that a binomial random variable is exceeding its mean. Our proof employs estimates on the mean absolute deviation and the tail conditional expectation of binomial random variables.

A new derivation of Golomb's limit formula for generating the $n$th$+1$ prime number is presented. The limit formula is derived by extracting $p_{n+1}$ from Euler's prime product representation of the Riemann zeta function $\zeta(s)$ in the…

Consider averages along the prime integers $ \mathbb P $ given by \begin{equation*} \mathcal{A}_N f (x) = N ^{-1} \sum_{ p \in \mathbb P \;:\; p\leq N} (\log p) f (x-p). \end{equation*} These averages satisfy a uniform scale-free $ \ell…

We show that there exists some $\delta > 0$ such that, for any set of integers $B$ with $B\cap[1,Y]\gg Y^{1-\delta}$ for all $Y \gg 1$, there are infinitely many primes of the form $a^2+b^2$ with $b\in B$. We prove a quasi-explicit formula…