Related papers: On an equation by primes with one Linnik prime

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that for every sufficiently large even integer $N$, the equation \begin{equation*}…

Let $\mathcal{R}$ denote the set of integers $n$ that can be represented as the sum $n = x^2 + y^2$ with $(x,y) = 1$. Let $a$ and $b$ be integers with $a>0$, $a \nmid b$. We show that for sufficiently large positive integer $N$ there are…

Let $\alpha$ be a real number such that $1< \alpha <2$ and let $x_0=x_0(\alpha)$ be a {\rm(}unique{\rm)} positive solution of the equation $$ x^{\alpha-1} -\frac{\pi}{e^2\sqrt{3}}x +1=0. $$ Then we prove that for each positive integer…

Let $a,b,c$ be positive integers. It is known that there are infinitely many positive integers not representated by $ax^2+by^2+cz^2$ with $x,y,z\in\mathbb Z$. In contrast, we conjecture that any natural number is represented by $\lfloor…

It is shown that the first $n$ prime numbers $p_1,...,p_n$ determine the next one by the recursion equation $$ p_{n+1} =\lim\limits_{s\to +\infty} [\prod\limits^n_{k=1} (1-\frac{1}{p^s_k}) \sum\limits^\infty_{j=1} \frac{1}{j^s} -1]^{-1/s}.…

This paper examines with elementary proofs some interesting properties of numbers in the binary quadratic form $a^2+ab+b^2$, where $a$ and $b$ are non-negative integers. Key findings of this paper are (i) a prime number $p$ can be…

Acquaah and Konyagin showed that if $N$ is an odd perfect number where $N= p_1^{a_1}p_2^{a_2} \cdots p_k^{a_k}$ where $p_1 < p_2 \cdots < p_k$ then one must have $p_k < 3^{1/3}N^{1/3}$. Using methods similar to theirs, we show that…

For a fixed positive integer $m$ and any partition $m = m_1 + m_2 + \cdots + m_e$ , there exists a sequence $\{n_{i}\}_{i=1}^{k}$ of positive integers such that $$m=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots+\frac{1}{n_{k}},$$ with the property…

Motivated by a 2001 problem of S\'ark\"ozy, we classify all situations of the integers $b,c,e$ and $f$ satisfying \begin{align*} \limsup_{n\rightarrow\infty}|d(\mathcal{A},bn+c)-d(\mathcal{A},en+f)|=\infty \end{align*} for any infinite…

We extend to Segal-Piatetski-Shapiro sequences previous results on the Luca-Schinzel question over integral valued polynomial sequences. Namely, we prove that for any real $c$ larger than $1$ the sequence $(\sum_{m\le n} \varphi(\lfloor m^c…

For $x>0$ let $\pi(x)$ denote the number of primes not exceeding $x$. For integers $a$ and $m>0$, we determine when there is an integer $n>1$ with $\pi(n)=(n+a)/m$. In particular, we show that for any integers $m>2$ and $a\le\lceil…

If $N = {p^k}{m^2}$ is an odd perfect number with special prime factor $p$, then it is proved that ${p^k} < (2/3){m^2}$. Numerical results on the abundancy indices $\frac{\sigma(p^k)}{p^k}$ and $\frac{\sigma(m^2)}{m^2}$, and the ratios…

We show that for any $\varepsilon > 0$, prime $q$ sufficiently large with respect to $1 / \varepsilon$ and residue class $(a,q) = 1$, there exist two integers $m, n \leq q^{5/2 + \varepsilon}$ with $m \equiv n \equiv a \pmod{q}$ such that…

Let $P_{r}$ denote an integer with at most $r$ prime factors counted with multiplicity. In this paper we prove that for some $\lambda < \frac{1}{12}$, the inequality $\{\sqrt{p}\}<p^{-\lambda}$ has infinitely many solutions in primes $p$…

Based on new explicit estimates for the prime counting function, we improve the currently known estimates for the particular sequence $C_n = np_n - \sum_{k \leq n}p_k$, $n \geq 1$, involving the prime numbers.

It is conjectured that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers $x,y$ and $z$, except for specific cases. We develop…

Hasanalizade [1] studied Deaconescu's conjecture for positive composite integer $n$. A positive composite integer $n\geq4$ is said to be a Deaconescu number if $S_2(n)\mid \phi(n)-1$. In this paper, we improve Hasanalizade's result by…

Let $\theta > 11/20$. We prove that every sufficiently large odd integer $n$ can be written as a sum of three primes $n = p_1 + p_2 + p_3$ with $|p_i - n/3| \leq n^{\theta}$ for $i\in\{1,2,3\}$.

Let $N$ be an odd perfect number and let $a$ be its third largest prime divisor, $b$ be the second largest prime divisor, and $c$ be its largest prime divisor. We discuss steps towards obtaining a non-trivial upper bound on $a$, as well as…

For any integer $l\geq 1$, let $p_1, p_2, \ldots, p_{l+2}$ be distinct prime numbers $\geq 5.$ For all real numbers $X>1,$ we let $N_{3,l}(X)$ denote the number of real quadratic fields $K$ whose absolute discriminant $d_K\leq X$ and $d_K$…