Related papers: Integers divisible by a large shifted prime

The degree of insulation of a prime $p$ is defined as the largest interval around it in which no other prime exists. Based on this, the $n$-th prime $p_{n}$ is said to be insulated if and only if its degree of insulation is higher than its…

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

Recall that a Stirling permutation is a permutation on the multiset $\{1,1,2,2,\ldots,n,n\}$ such that any numbers appearing between repeated values of $i$ must be greater than $i$. We call a Stirling permutation ``flattened'' if the…

The pentagonal number theorem is extended to the sequence of the number of integer partitions with all parts equal. The new pentagonal number theorem implies that the distribution of the primes is just a specific detail of the application…

For two sets $A$ and $M$ of positive integers and for a positive integer $n$, let $p(n,A,M)$ denote the number of partitions of $n$ with parts in $A$ and multiplicities in $M$, that is, the number of representations of $n$ in the form…

Denote by $\mathbb{N}$ and $\mathbb{P}$ the set of all positive integers and prime numbers, respectively. Let $\mathbb{P}=\{p_1<p_2<\dots <p_n<\dots\}$, where $p_n$ is the $n$-th prime number. For $k\in\mathbb{N}$ we recursively define…

Let $x,h$ and $Q$ be three parameters. We show that, for most moduli $q\le Q$ and for most positive real numbers $y\le x$, every reduced arithmetic progression $a\mod q$ has approximately the expected number of primes $p$ from the interval…

We show that if a big set of integer points in [0,N]^d, d>1, occupies few residue classes mod p for many primes p, then it must essentially lie in the solution set of some polynomial equation of low degree. This answers a question of…

The theorem presented in this paper allows the creation of large prime numbers (of order up to o(n^2)) given a table of all primes up to n.

Motivated by a question of V. Bergelson and F. K. Richter (2017), we obtain asymptotic formulas for the number of relatively prime tuples composed of positive integers $n\le N$ and integer parts of polynomials evaluated at $n$. The error…

Let S(n,k) denote the Stirling numbers of the second kind. We prove that the p-adic limit of S(p^e a + c, p^e b + d) as e goes to infinity exists for all integers a, b, c, and d. We call the limiting p-adic integer S(p^\infty a + c,…

Let $f(x)$ be a nonconstant polynomial with integer coefficients and nonzero discriminant. We study the distribution modulo primes of the set of squarefree integers $d$ such that the curve $dy^2=f(x)$ has a nontrivial rational or integral…

We study number theoretic properties of the map $x \mapsto x^{x} \mod{p}$, where $x \in \{1,2,\ldots,p-1\}$, and improve on some recent upper bounds, due to Kurlberg, Luca, and Shparlinski, on the number of primes $p < N$ for which the map…

We present a general theorem characterizing the relationship between the prime base $p$ representations of non-negative integers $x$ and their positive integer powers, $x^n$. For any positive integer $l$, the theorem establishes the…

For two odd primes $p$ and $q$ such that $p<q$, let $A(p,q):=(a_k)_{k=1}^{\infty}$ be the arithmetic progression whose $k$th term is given by $a_k=(k-1)(q-p)+p$ (i.e., with $a_1=p$ and $a_2=q$). Here we conjecture that for every positive…

Ferrers diagrams are used to visually represent integer partitions. We describe a way to use Ferrers diagrams to uniquely represent integers in terms of their prime factors. This leads to a lower bound on the number of primes less than a…

We establish asymptotic upper bounds on the number of zeros modulo $p$ of certain polynomials with integer coefficients, with $p$ prime numbers arbitrarily large. The polynomials we consider have degree of size $p$ and are obtained by…

Inspired by the proof of the Bertrand postulate given by P. Erd\H{o}S, we carefully examine and solve one less usual inequality in positive integers which could help to find an arithmetically pure proof that for every positive integer…

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…

For each of the functions $f \in \{\phi, \sigma, \omega, \tau\}$ and every natural number $k$, we show that there are infinitely many solutions to the inequalities $f(p_n-1) < f(p_{n+1}-1) < \dots < f(p_{n+k}-1)$, and similarly for…