Related papers: Shifted-prime divisors
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…
A unitary divisor $c$ of a positive integer $n$ is a positive divisor of $n$ that is relatively prime to $\displaystyle{\frac{n}{c}}$. For any integer $k$, the function $\sigma_k^*$ is a multiplicative arithmetic function defined so that…
We prove lower bounds of the form $\gg N/(\log N)^{3/2}$ for the number of primes up to $N$ primitively represented by a shifted positive definite integral binary quadratic form, and under the additional condition that primes are from an…
A number $n$ is practical if every integer in $[1,n]$ can be expressed as a subset sum of the positive divisors of $n$. We consider the distribution of practical numbers that are also shifted primes, improving a theorem of Guo and…
We explicitly describe the splitting of odd integral primes in the radical extension $\mathbb{Q}(\sqrt[n]{a})$, where $x^n-a$ is an irreducible polynomial in $\mathbb{Z}[x]$. Our motivation is to classify common index divisors, the primes…
We show that there are infinitely many primes $p$ such that not only does $p + 2$ have at most two prime factors, but $p + 6$ also has a bounded number of prime divisors. This refines the well known result of Chen.
We present the first fixed-length elementary closed-form expressions for the prime-counting function, $\pi(n)$, and the $n$-th prime number, $p(n)$. These expressions are arithmetic terms, requiring only a finite and fixed number of…
We study some divisibility properties of multiperfect numbers. Our main result is: if $N=p_1^{\alpha_1}... p_s^{\alpha_s} q_1^{2\beta_1}... q_t^{2\beta_t}$ with $\beta_1, ..., \beta_t$ in some finite set S satisfies…
Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense…
The Prime Number Theorem states that the number of primes in $\{1,\ldots,x\}$, denoted $\pi(x)$, is approximately $\frac{x}{\ln(x)}$. In this paper, we investigate the distribution of primes for domains other than $\N$. First we look at…
We investigate the first and second moments of shifted convolutions of the generalised divisor function $d_3(n)$.
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…
We show that for integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the normal order of the number of prime factors is $C \log \log n$, where $C=(1-e^{-\gamma})^{-1} = 2.280...$ and $\gamma$ is…
We call an integer N>1 primover to base a if it either prime or overpseudoprime to base a. We prove, in particular, that every Fermat number is primover to base 2. We also indicate a simple process of receiving of primover divisors of…
We prove that the upper bound for the van der Corput property of the set of shifted primes is O((log n)^{-1+o(1)}), giving an answer to a problem considered by Ruzsa and Montgomery for the set of shifted primes p-1. We construct normed…
Euler states without proof statements about the form of prime divisors of numbers of the form aa+Nbb. See Ed Sandifer's How Euler Did It, ``Factors of Forms'', December 2005 at http://www.maa.org/news/howeulerdidit.html for a summary of the…
We study primitive divisors of terms of the sequence P_n=n^2+b, for a fixed integer b which is not a negative square. It seems likely that the number of terms with a primitive divisor has a natural density. This seems to be a difficult…
The purpose of this note is to report on the discovery of the primes of the form $p=1+n!\sum n$, for some natural numbers $n>0$. The number of digits in the prime p are approximately equal to $\lfloor log_{10}(1+n!\sum n)\rceil+1$.
Let $\Omega(n)$ denote the number of prime factors of $n$. We show that for any bounded $f\colon\mathbb{N}\to\mathbb{C}$ one has \[ \frac{1}{N}\sum_{n=1}^N\, f(\Omega(n)+1)=\frac{1}{N}\sum_{n=1}^N\, f(\Omega(n))+\mathrm{o}_{N\to\infty}(1).…
In the number $373$ all subwords ($3$, $7$, $37$, $73$, and $373$) are prime. Similarly, in $9719$ all subwords are divisible by at most one prime. And similarly again in $7319797913$ all subwords are divisible by at most two primes. These…