Related papers: On an logarithmic equation by primes

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…

We show that while the number of coprime compositions of a positive integer $n$ into $k$ parts can be expressed as a $\mathbb{Q}$-linear combinations of the Jordan totient functions, this is never possible for the coprime partitions of $n$…

Define {\em the Liouville function for $A$}, a subset of the primes $P$, by $\lambda_{A}(n) =(-1)^{\Omega_A(n)}$ where $\Omega_A(n)$ is the number of prime factors of $n$ coming from $A$ counting multiplicity. For the traditional Liouville…

We prove an asymptotic formula for the number of primes of the shape $a^2 +p^4$, thereby refining the well known work of Friedlander and Iwaniec. Along the way, we prove a result on equidistribution of primes up to $x$, in which the moduli…

For $n \geq 3,$ let $ p_n $ denote the $n^{\rm th}$ prime number. Let $[ \; ]$ denote the floor or greatest integer function. For a positive integer $m,$ let $\pi_2(m)$ denote the number of twin primes not exceeding $m.$ The twin prime…

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…

An algebraic integer is said large if all its real or complex embeddings have absolute value larger than $1$. An integral ideal is said \emph{large} if it admits a large generator. We investigate the notion of largeness, relating it to some…

We investigate the average number of representations of a positive integer as the sum of $k + 1$ perfect $k$-th powers of primes. We extend recent results of Languasco and the last Author, which dealt with the case $k = 2$ [6] and $k = 3$…

When k > 1 and s is sufficiently large in terms of k, we derive an explicit multi-term asymptotic expansion for the number of representations of a large natural number as the sum of s positive integral k-th powers.

Let $[\, x\,]$ denote the integer part of a real number $x$. Assume that $\lambda_1,\lambda_2,\lambda_3$ are nonzero real numbers, not all of the same sign, that $\lambda_1/\lambda_2$ is irrational, and that $\eta$ is real. Let…

Let $p$ be a prime, $\varepsilon>0$ and $0<L+1<L+N < p$. We prove that if $p^{1/2+\varepsilon}< N <p^{1-\varepsilon}$, then $$ \#\{n!\!\!\! \pmod p;\,\, L+1\le n\le L+N\} > c (N\log N)^{1/2},\,\, c=c(\varepsilon)>0. $$ We use this bound to…

In this paper, we consider representations of integers as sums of at most four distinct $m$-gonal numbers (allowing a fixed number of repeats of each polygonal number occurring in the sum). We show that the number of such representations…

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 $F_1,\ldots,F_R$ be homogeneous polynomials of degree $d\ge 2$ with integer coefficients in $n$ variables, and let $\mathbf{F}=(F_1,\ldots,F_R)$. Suppose that $F_1,\ldots,F_R$ is a non-singular system and $n\ge 4^{d+2}d^2R^5$. We prove…

Building on techniques recently introduced by the second author, and further developed by the first author, we show that a positive integer $N$ may be rigorously and deterministically factored into primes in at most \[ O\left( \frac{N^{1/5}…

For a fixed prime $p$, let $e_p(n!)$ denote the order of $p$ in the prime factorization of $n!$. Chen and Liu (2007) asked whether for any fixed $m$, one has $\{e_p(n^2!) \bmod m:\; n\in\mathbb{Z}\}=\mathbb{Z}_m$ and $\{e_p(q!) \bmod m:\; q…

We give a deterministic algorithm that very quickly proves the primality or compositeness of the integers N in a certain sequence, using an elliptic curve E/Q with complex multiplication by the ring of integers of Q(sqrt(-7)). The algorithm…

For a natural number $k>1$, let $f_k(n)$ denote the number of distinct representations of a natural number $n$ of the form $p^k+q^k$ for primes $p,q$. We prove that, for all $k>1$, $$\limsup_{n\to\infty}f_k(n)=\infty.$$ This positively…

This note presents a result on the maximal prime gap of the form p_(n+1) - p_n <= C(log p_n)^(1+e), where C > 0 is a constant, for any arbitrarily small real number e > 0, and all sufficiently large integer n > n_0. Equivalently, the result…

For "almost all" sufficiently large $N,$ satisfying necessary congruence conditions and $k\geq 2$, we show that there is an {\bf asymptotic formula} for the number of solutions of the equation \begin{align*} \begin{split}…