Related papers: Integers divisible by a large shifted prime

Every odd prime number p can be written in exactly (p + 1)/2 ways as a sum ab + cd with min(a, b) > max(c, d) of two ordered products. This gives a new proof Fermat's Theorem expressing primes of the form 1 + 4N as sums of two squares 1 .

A natural number $n$ is called semi-prime if it is a product of two primes or a square of a prime. We denote $\mathbb{P}_2$ the set of all semi-primes. Our goal is to prove that for fixed integer number $a$ and sufficiently large $x$ the…

Suppose that $A \subset \{1,\dots, N\}$ has no two elements differing by $p-1$, $p$ prime. Then $|A| \ll N^{1 - c}$.

In Pacific J. Math. 292 (2018), 223-238, Shareshian and Woodroofe asked if for every positive integer $n$ there exist primes $p$ and $q$ such that, for all integers $k$ with $1 \leq k \leq n-1$, the binomial coefficient $\binom{n}{k}$ is…

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…

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…

We develop an analog for shifted primes of the Kubilius model of prime factors of integers. We prove a total variation distance estimate for the difference between the model and actual prime factors of shifted primes, and apply it to show…

For $n=1,2,3,\ldots$ let $S_n$ be the sum of the first $n$ primes. We mainly show that the sequence $a_n=\root n\of{S_n/n}\ (n=1,2,3,\ldots)$ is strictly decreasing, and moreover the sequence $a_{n+1}/a_n\ (n=10,11,\ldots)$ is strictly…

The problem of N-digit sets all permutations of which give primes is discussed. Such sets may include only digits 1, 3, 7 and 9, and none of 0, 2, 5, 4, 6, 8. Direct calculations show that such full-permutation digit sets occur at N = 1, 2,…

We prove that there are infinitely many integers $n$ such that $n$ and $n+1$ have the same number of distinct prime divisors.

Let $p$ be a prime number, and $h$ a positive integer such that $\gcd(p,h)=1$. We prove, without invoking Dirichlet's theorem, that the arithmetic progression $p\left(\mathbf{N}\cup \{0\}\right)+h$ contains infinitely many prime numbers.…

In this work we show that the prime distribution is deterministic. Indeed the set of prime numbers P can be expressed in terms of two subsets of N using three specific selection rules, acting on two sets of prime candidates. The prime…

We introduce $p$-derivations and give a few basic ways in which they act like derivatives by numbers.

Let $N$ be a positive integer and let $S_N$ be the set of polynomials with integer coefficients, degree less than $N$, and minimal positive integral over $[0,1]$. D. Bazzanella initiated the study of $S_N$ because of its relation to the…

For all integers $n \geq k \geq 1$, define $H(n,k) := \sum 1 / (i_1 \cdots i_k)$, where the sum is extended over all positive integers $i_1 < \cdots < i_k \leq n$. These quantities are closely related to the Stirling numbers of the first…

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…

It is well-known that for any distinct positive integers $k$ and $n$, the numbers $2^{2^k}+1$ and $2^{2^n}+1$ are relatively prime. In this paper we consider the situation when 1 is replaced by some positive integer $d>1$

We prove new results on the additive theory of reversed primes $\overleftarrow{p}$; that is, primes $p$ which are written backwards in a fixed base $b\geq 2$. In particular, we study a variant of Goldbach's conjecture, looking at…

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…

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 $d_K$…