Related papers: On some floor function sets

We treat the functions $\star^k:{\mathbf N}\rightarrow{\mathbf N}$ where $\star:x\mapsto \star x := x(x+1)$. The set $\{\star^k x+1: \{x,k\}\subseteq{\mathbf N}\}$ is pairwise coprime; so, the set ${\mathbf P}$ of primes is infinite. Our…

Recently there has been several works estimating the number of $n\times n$ matrices with elements from some finite sets $\mathcal X$ of arithmetic interest and of a given determinant. Typically such results are compared with the trivial…

Numerical study of the distribution of the Riemann zeros differences following the work [1] shows the significance of the function for which the prime sum expression is proposed. Computational results related to this definition explored…

Let M be the ideal of first category subsets of R and non(M)=min{card X: X \subseteq R, X \not\in M}. We consider families \Phi of sequences converging to \infty, with the property that for every open set U \subseteq R that is unbounded…

Let $G$ be a graph. A dominating set $D\subseteq V(G)$ is a super dominating set if for every vertex $x\in V(G) \setminus D$ there exists $y\in D$ such that $N_G(y)\cap (V(G)\setminus D)) = \{x\}$. The cardinality of a smallest super…

Let $(p_n)$ denote the sequence of prime numbers, with $2=p_1<p_2<\ldots$. We demonstrate the existence of an irrational number $\alpha$ having the property that the sequence $(\alpha p_n)$ is not well-distributed modulo $1$.

Motivated by recent results, we study sums of the form $S_f(x) = \sum_{n\leq x} f\left(\left\lfloor\frac{x}{n}\right\rfloor \right)$, where $f$ is an arithmetic function and $\left\lfloor\cdot\right\rfloor$ denotes the greatest integer…

Different questions lead to the same class of functions from natural integers to integers: those which have integral difference ratios, i.e. verifying $f(a)-f(b)\equiv0 \pmod {(a-b)}$ for all $a>b$. We characterize this class of functions…

Let $p_n$ be $n$th prime, and let $(S_n)_{n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum_{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding…

We consider the summatory function of the number of prime factors for integers $\leq x$ over arithmetic progressions. Numerical experiments suggest that some arithmetic progressions consist more number of prime factors than others. Greg…

Let $f(x)=x^n+a_{n-1}x^{n-1}+\dots+a_0$ be an irreducible polynomial with integer coefficients. For a prime $p$ for which $f(x)$ is fully splitting modulo $ p$, we consider $n$ roots $r_i$ of $f(x)\equiv 0\bmod p$ with $0 \le r_1\le\dots\le…

Let $[\, \cdot\,]$ be the floor function. In this paper we show that when $1<c<\frac{3849}{3334}$, then there exist infinitely many prime numbers of the form $[n^c]$, where $n$ is square-free.

Mills showed that there exists a constant $A$ such that $\lfloor{A^{3^n}}\rfloor$ is prime for every positive integer $n$. Kuipers and Ansari generalized this result to $\lfloor{A^{c^n}}\rfloor$ where $c\in\mathbb{R}$ and $c\geq 2.106$. The…

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…

We identify pairs of positive integers $(t, d)$ with the property that the integer sequence with general term $\lfloor{n^t/d\rfloor}$ contains at most finitely many primes.

For a fixed positive integer d, we show the existence of the limiting gap distribution measure for the sets of Farey fractions a/q of order Q with a not divisible by d, and respectively with q relatively prime with d, as Q tends to…

We classify nilpotent pre-Lie rings of cardinality $p^4$ and thereby braces of the same cardinality, for a sufficiently large prime $p$. It has been shown that nilpotent pre-Lie rings of cardinality $p^n$ correspond to strongly nilpotent…

The probability of finding a prime multiplet, i.e., a sequence of primes $p$ and $p+a_i$, $i=1... m$, being all primes where $p$ is some prime less than the integer $n$ is naively $1/log(n)^{m+1}$. It is shown that, in reality, it is…

In this paper we discuss a method to express the Prime counting function as a "sum" over Non-trivial zeros of Riemann Zeta function, using techniques from Analytic Number Theory, also we apply our results to the sum over primes of any…

In this short note we prove the following result: If a completely multiplicative function $f:\mathbb{N}\to[-1,1]$ is small on average in the sense that $\sum_{n\leq x}f(n)\ll x^{1-\delta}$, for some $\delta>0$, and if the Dirichlet series…