Related papers: An asymptotic version of the prime power conjectur…

In a recent work \cite{key-11}, A. Fish proved that if $E_{1}$ and $E_{2}$ are two subsets of $\mathbb{Z}$ of positive upper Banach density, then there exists $k\in\mathbb{Z}$ such that…

We study pairs and m--tuples of compositions of a positive integer n with parts restricted to a subset P of positive integers. We obtain some exact enumeration results for the number of tuples of such compositions having the same number of…

Given positive integers $a_1,\ldots,a_k$, we prove that the set of primes $p$ such that $p \not\equiv 1 \bmod{a_i}$ for $i=1,\ldots,k$ admits asymptotic density relative to the set of all primes which is at least $\prod_{i=1}^k…

Let $p/q$ ($p, q \in \mathbb{N}^*$) be a positive rational number such that $p > q^2$. We show that for any $\epsilon > 0$, there exists a set $A(\epsilon) \subset [0, 1[$, with finite border and with Lebesgue measure $< \epsilon$, for…

The twin prime conjecture asserts that there are infinitely many pairs of primes that differ by two. While recent advances have improved our understanding of bounded prime gaps, the conjecture remains unresolved. This paper refines the…

In this paper, we improve the results in the author's previous paper \cite{Usu22}, which deals with the quantitative problem on Littlewood's conjecture. We show that, for any $0<\gamma<1$, any $(\alpha,\beta)\in\mathbb{R}^2$ except on a set…

We study the set $\mathcal{S}$ of odd positive integers $n$ with the property ${2n}/{\sigma(n)} - 1 = 1/x$, for positive integer $x$, i.e., the set that relates to odd perfect and odd "spoof perfect" numbers. As a consequence, we find that…

Generalizing the concept of a perfect number, Sloane's sequences of integers A083207 lists the sequence of integers $n$ with the property: the positive factors of $n$ can be partitioned into two disjoint parts so that the sums of the two…

Let $W\subseteq \mathbb{Z}$ be a non-empty subset of the integers. A nonempty set $C\subseteq \mathbb{Z}$ is said to be an asymptotic complement to $W$ if $W+C$ contains almost all the integers except a set of finite size. $C$ is said to be…

We prove the Ribenboim hypothesis, which states that if, starting from some integer $N$, consecutive prime numbers $p_ {n}$, $p_{n+1}$ satisfy the inequality $\sqrt {p_ {n+1}}-\sqrt{p_{n}} <1$, then the Landau problem # 4 (1912) has a…

Let $F$ and $G$ be linear recurrences over a number field $\mathbb{K}$, and let $\mathfrak{R}$ be a finitely generated subring of $\mathbb{K}$. Furthermore, let $\mathcal{N}$ be the set of positive integers $n$ such that $G(n) \neq 0$ and…

Let $n\in\mathbb{Z}^+$. Is it true that every sequence of $n$ consecutive integers greater than $n^2$ and smaller than $(n+1)^2$ contains at least one prime number? In this paper we show that this is actually the case for every $n \leq…

We prove that a positive proportion of the intervals of any fixed scalar multiple of $\log(X)$ in the dyadic interval $[X,2X]$ contain a prime number. We also show that a positive proportion of the congruence classes modulo $q$ contain a…

Let $\epsilon$ be a fixed positive quantity, $m$ be a large integer, $x_j$ denote integer variables. We prove that for any positive integers $N_1,N_2,N_3$ with $N_1N_2N_3>m^{1+\epsilon},$ the set $$ \{x_1x_2x_3 \pmod m: \quad x_j\in [1,N_j]…

For a finite set of non-zero natural numbers that contains at least one element different from 1 and the least common multiple of any of its subsets, there exists a subset of at least half of its members which has a common divisor larger…

Let $K$ be a number field, $k\geq 2$ an integer, $(K^*)^k$ the $k$-fold direct product of $K^*$ with coordinatewise multiplication, and $\Gamma$ a finitely generated subgroup of rank $r$ of $(K^*)^k$. Further, let $H(\alpha )$ denote the…

Let $h,k \ge 2$ be integers. We say a set $A$ of positive integers is an asymptotic basis of order $k$ if every large enough positive integer can be represented as the sum of $k$ terms from $A$. A set of positive integers $A$ is called…

Let $x$ be a positive integer. We give an asymptotic formula for the number of primes in the set $\{\fl{x/n}, 1 \le n \le x\}$ and give some related results.

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,…

Let $p$ be a sufficiently large prime number, $r$ be any given positive integer. Suppose that $a_1,\,\dots,\,a_r$ are pairwise distinct and not zero modulo $p$. Let $N(a_1,\,\dots,\,a_r;\,p)$ denote the number of…