Related papers: Multiplicative Functions on Shifted Primes

We consider the integers having the property of reversing when multiplied by a specific integer k. First, we proved that k should be either 1, 4 or 9. Second, we classify these integers as (10, 1)- reverse multiples, (10, 4)- reverse…

We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.

We deduce an asymptotic formula with error term for the sum $\sum_{n_1,\ldots,n_k \le x} f([n_1,\ldots, n_k])$, where $[n_1,\ldots, n_k]$ stands for the least common multiple of the positive integers $n_1,\ldots, n_k$ ($k\ge 2$) and $f$…

We present a novel conjecture concerning the additive representation of natural numbers using prime powers. Based on extensive computational verification, we conjecture that every integer n > 23 can be expressed as a sum of at most five…

The function f:X -> Y is called k-monotonically increasing if there is a partition X = X_1 U ... U X_k such that f|X_i : X_i -> Y is monotonically increasing for i=1,...,k. It is proved that a one-to-one function f:N -> N is k-monotonically…

It is known that there are infinitely-many prime numbers which take the form of a polynomial of degree one with integer coefficients, this is Dirichlet's theorem. We use an elementary sieving argument together with bounds on the prime…

The sequence $$ \mathbb{P}^{(c)}=(\lfloor p^c \rfloor)_{p\in \mathbb{P}}\quad (c>0,c\notin \mathbb{N}), $$ is an important subsequence of the well-known Piatetski-Shapiro sequence, where $\mathbb{P}$ is the set of prime numbers and $\lfloor…

A prime number $p$ is said to be a Wolstenholme prime if it satisfies the congruence ${2p-1\choose p-1} \equiv 1 \,\,(\bmod{\,\,p^4})$. For such a prime $p$, we establish the expression for ${2p-1\choose p-1}\,\,(\bmod{\,\,p^8})$ given in…

For a base $b\geq 2$ and a set of digits $\mathcal{A}\subset \{0,...,b-1\}$, let $\mathcal{P}$ denote the set of prime numbers with digits restricted to $\mathcal{A}$, when written in base-$b$. We prove that if $A\subset \mathbb{N}$ has…

Let $A^{(n)}_{l;k}\subset S_n$ denote the event that the set of $l$ consecutive numbers $\{k,k+1,\cdots, k+l-1\}$ appear in a set of $l$ consecutive positions. Let $p=\{p_j\}_{j=1}^\infty$ be a distribution on $\mathbb{N}$ with $p_j>0$. Let…

We say a polynomial f having integer coefficients is strongly coefficient convex if the set of coefficients of f consists of consecutive integers only. We establish various results suggesting that the divisors of x^n-1 with integer…

An integer is said to be $y$-friable if its greatest prime factor P(n) is less than $y$. In this paper, we study numbers of the shape $n-1$ when $P(n)\leq y$ and $n\leq x$. One expects that, statistically, their multiplicative behaviour…

Let $X$ be a scheme of finite type over $\mathbf{Z}$. For $p \in \mathcal{P}$ the set of prime numbers, let $N_{X}(p)$ be the number of $\mathbf{F}_{p}$-points of $X/\mathbf{F}_{p}$. For fixed $n\geq 1$ and $a_{1}, \ldots, a_{n} \in…

For a set of primes $\mathcal{P}$, let $\Psi(x, \mathcal{P})$ be the number of positive integers $n \leq x$ all of whose prime factors lie in $\mathcal{P}$. In this paper we classify the sets of primes $\mathcal{P}$ such that $\Psi(x,…

Let $\mathbb{F}_p$ be a finite field of prime order $p$ and let $A \subset \mathbb{F}_p$ be a subset. In the dense regime when $|A| \geq \alpha p$ for some $\alpha \in (0,1)$, we determine the optimal constant $f(\alpha)$ in the inequality…

Let $[\, \cdot\,]$ be the floor function and $\|x\|$ denotes the distance from $x$ to the nearest integer. In this paper we show that whenever $\alpha$ is irrational and $\beta$ is real then for any fixed $1<c<12/11$ there exist infinitely…

We determine the order of magnitude of H^{(k+1)}(x,\vec{y},2\vec{y}), the number of integers up to x that are divisible by a product d_1...d_k with y_i<d_i\le 2y_i, when the numbers \log y_1,...,\log y_k have the same order of magnitude and…

We call a set of positive integers closed under taking unitary divisors a unitary ideal. It can be regarded as a simplicial complex. Moreover, a multiplicative arithmetical function on such a set corresponds to a function on the simplicial…

Let $p$ be a prime number. We say that a positive integer $n$ is a Sylow $p$-number if there exists a finite group having exactly $n$ Sylow $p$-subgroups. When $p=2$, every odd integer is a Sylow $2$-number. In contrast, when $p$ is odd,…

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…