Related papers: A large integer is a sum of two prime avoiding num…

Let $\{f_n\}$ be the Fibonacci sequence. For any positive integer $n$, let $r(n)$ be the number of solutions of $n=p+f_{k_1^{2}} +f_{k_{2}^{2}}$, where $p$ is a prime and $k_1, k_2$ are nonnegative integers with $k_1\le k_2$. In this paper,…

Let $p$ be a prime. We define $S(p)$ the smallest number $k$ such that every positive integer is a sum of at most $k$ squares of integers that are not divisible by $p$. In this article, we prove that $S(2)=10$, $S(3)=6$, $S(5)=5$, and…

In this paper, we find all integers $x$ such that $x^{2}-1$ has only prime factors smaller than 100. This gives some interesting numerical corollaries. For example, for any positive integer $n$ we can find the largest positive integer $x$…

Let $\delta > 1/2$. We prove that if $A$ is a subset of the primes such that the relative density of $A$ in every reduced residue class is at least $\delta$, then almost all even integers can be written as the sum of two primes in $A$. The…

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…

A Toda prime of an integer $n$ is an odd prime $p$ such that $4n=(p-1)k$ with $k$ coprime to $p$. We conjecture that every positive integer admits at least two Toda primes. We give a partial proof that every positive integer admits at least…

A positive integer $n$ is called an abundant number if $\sigma (n)\ge 2n$, where $\sigma (n)$ is the sum of all positive divisors of $n$. Let $E(x)$ be the largest number of consecutive abundant numbers not exceeding $x$. In 1935, P. Erd\H…

For an $n$-bit positive integer $a$ written in binary as $$ a = \sum_{j=0}^{n-1} \varepsilon_{j}(a) \,2^j $$ where, $\varepsilon_j(a) \in \{0,1\}$, $j\in\{0, \ldots, n-1\}$, $\varepsilon_{n-1}(a)=1$, let us define $$ \overleftarrow{a} =…

Let p be any prime, and $p^(\nu_p(n!))$ the maximal power of $p$ dividing $n!$. It is proved that there exists a positive integer $n_0$, which depends only on $p$, such that $q^(\nu_q(n!)) < p^(\nu_p(n!))$ for all $n \ge n_0$ and all primes…

Let $p_{1}<p_2<... <p_{\nu}<...$ be the sequence of prime numbers and let $m$ be a positive integer. We give a strong asymptotic formula for the distribution of the set of integers having prime factorizations of the form…

Let $N$ denotes a sufficiently large even integer, $p$ denotes a prime and $P_{r}$ denotes an integer with at most $r$ prime factors. In this paper, we study the solutions of the equation $N-p=P_3$ and consider two special cases where $p$…

We prove there exist infinitely many odd integers $n$ for which there exists a pair of positive divisors $d_1, d_2>1$ of $(n^2+1)/2$ such that $$d_1+d_2=\delta n+(\delta+2).$$ We prove the similar result for $\varepsilon=\delta-2$ and…

Let $Q$ be a set of primes with relative density $\delta$. We count integers in $[1,x]$ with prime factors all in $Q$ that also have a divisor in $(y,2y]$. We establish the order of magnitude for all $\delta \in (0,1]$. This generalizes the…

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 nonempty finite set of relatively prime positive integers, and let p_A(n) denote the number of partitions of n with parts in A. An elementary arithmetic argument is used to obtain an asymptotic formula for p_A(n).

We prove that if $x$ is large enough, namely $x\ge x_0$, then there exists a prime between $x(1- \Delta^{-1})$ and $x$, where $\Delta$ is an effective constant computed in terms of $x_0$. This improves some previous results of Ramar\'e and…

We estimate from below the lower density of the set of prime numbers p such that p-1 has a prime factor of size at least p^c, where c lies in between 1/4 and 1/2. We also establish upper and lower bounds on the counting function of the set…

For a nonempty finite set $A$ of positive integers, let $\gcd\left(A\right)$ denote the greatest common divisor of the elements of $A$. Let $f\left(n\right)$ and $\Phi\left(n\right)$ denote, respectively, the number of subsets $A$ of…

In this paper, it is established that every sufficiently large positive integer $n$ subject to $n\equiv0\pmod2$ can be represented as a sum of one square of prime and seventeen fifth powers of primes, which gives an enhancement upon 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…