Related papers: Small gaps between products of two primes

In the first paper under this title (1977), the first author utilized a duality identity between the largest and smallest prime factors involving the Moebius function, to establish the following result as a consequence of the Prime Number…

Let $Q$ be a positive-definite quaternary quadratic form with prime discriminant. We give an explicit lower bound on the number of representations of a positive integer $n$ by $Q$. This problem is connected with deriving an upper bound on…

Every natural number greater than $2$ can be written as the sum of a prime and a square-free number, and recent work has imposed additional divisibility conditions on the square-free number. We overcome limitations in these works to prove…

We discuss $Q(n)$, the number of ways a given integer $n$ may be written as a sum of distinct primes, and study its asymptotic form $Q_{as}(n)$ valid in the limit $n\to\infty$. We obtain $Q_{as}(n)$ by Laplace inverting the fermionic…

We first prove that if $a$ has a prime factor not dividing $b$ then there are infinitely many positive integers $n$ such that $\binom {an+bn} {an}$ is not divisible by $bn+1$. This confirms a recent conjecture of Z.-W. Sun. Moreover, we…

Let $\sigma(n)$ be the sum of the positive divisors of $n$. A positive integer $n$ is said to be $2$-near perfect when $\sigma(n)=2n+d_1+d_2$, where $d_1$ and $d_2$ are distinct positive divisors of $n$. We show that there are no odd…

It is sufficient to prove that there is an excess of prime factors in the product of repunits with odd prime bases defined by the sum of divisors of the integer $N=(4k+1)^{4m+1}\prod_{i=1}^\ell ~ q_i^{2\alpha_i}$ to establish that there do…

Let $\mathcal{A}=\{a_{n}\}_{n=1}^{\infty}$ and $\mathcal{B}=\{b_{n}\}_{n=1}^{\infty}$ be two sequences of positive integers (not necessarily distinct). Under some restrictions on $\mathcal{A}$ and $\mathcal{B}$, we obtain a lower bound for…

We prove that there are infinitely many solutions of $$ |\lambda_0+\lambda_1p+\lambda_2P_r|<p^{-\tau}, $$ where $r=3,$ $\tau=\frac1{118}$, and $\lambda_0$ is an arbitrary real number and $\lambda_1,\lambda_2\in\BR$ with $\lambda_2\neq0$ and…

For each integer $b \geq 3$ and every $x \geq 1$, let $\mathcal{N}_{b,0}(x)$ be the set of positive integers $n \leq x$ which are divisible by the product of their nonzero base $b$ digits. We prove bounds of the form $x^{\rho_{b,0} + o(1)}…

It is well-known that for any non-constant polynomial $P$ with integer coefficients the sequence $(P(n))_{ n\in \mathbb N}$ has the property that there are infinitely many prime numbers dividing at least one term of this sequence.…

For a positive integer $n$ let $\mathfrak{P}_n=\prod_{s_p(n)\ge p} p,$ where $p$ runs over all primes and $s_p(n)$ is the sum of the base $p$ digits of $n$. For all $n$ we prove that $\mathfrak{P}_n$ is divisible by all "small" primes with…

We show that there exists pairs of consecutive primes less than $x$ whose difference is larger than $t(1+o(1))(\log{x})(\log\log{x})(\log\log\log\log{x})(\log\log\log{x})^{-2}$ for any fixed $t$. Our proof works by incorporating recent…

Every odd prime number p can be written in exactly (p + 1)/2 ways as a sum ab+cd of two ordered products ab and cd such that min(a, b) > max(c, d). An easy corollary is a proof of Fermat's Theorem expressing primes in 1 + 4N as sums of two…

Erd\H{o}s asked whether there are infinitely many finite sets of distinct primes $p_1<\cdots<p_k$ and positive integers $m$ such that \begin{equation}\label{eq:erdos-original} \frac1{p_1}+\cdots+\frac1{p_k}=1-\frac1m. \end{equation} This is…

Let $p_1 = 2, p_2 = 3,...$ be the sequence of all primes. Let $\epsilon$ be an arbitrarily small but fixed positive number, and fix a coprime pair of integers $q \ge 3$ and $a$. We will establish a lower bound for the number of primes…

By Maynard's theorem and the subsequent improvements by the Polymath Project, there exists a positive integer $b\leq 246$ such that there are infinitely many primes $p$ such that $p+b$ is also prime. Let $P_1,...,P_t\in \mathbb{Z}[y]$ with…

Twin prime number problem is mainly the structure of the twin prime numbers and whether there are infinitely many prime twins group. In this paper, by constructing a special cluster number set(see formula(2.3)in the paper), proves that the…

A linear combination $aT_r(m)+bT_s(n)$ of an \mbox{$r$-gonal} number $T_r(m)$ and an $s$-gonal number $T_s(n)$ with mutually coprime positive integer coefficients $a$ and $b$ produces infinitely many primes as $m$ and~$n$ varies over the…

Is it true that for all $n\geq k\geq 2$ there exists a prime number between $kn$ and $(k+1)n$? In this paper we show that there is always a prime number between $4n$ and $5n$ for all $n>2$. We also show there are at least seven prime…