Related papers: A note on OEIS sequence A111384

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.

For two coprime positive integers $a$ and $b$,let $\pi^* (a, b)$ be the number of primes that cannot be represented as $au+bv$, where $u$ and $v$ are nonnegative integers. It is clear that $\pi^* (a, b)\le \pi (ab-a-b)$, where $\pi (x)$…

Mills proved that there exists a real constant $A>1$ such that for all $n\in \mathbb{N}$ the values $\lfloor A^{3^n}\rfloor$ are prime numbers. No explicit value of $A$ is known, but assuming the Riemann hypothesis one can choose $A=…

It is shown that the first $n$ prime numbers $p_1,...,p_n$ determine the next one by the recursion equation $$ p_{n+1} =\lim\limits_{s\to +\infty} [\prod\limits^n_{k=1} (1-\frac{1}{p^s_k}) \sum\limits^\infty_{j=1} \frac{1}{j^s} -1]^{-1/s}.…

The starting point of our paper is Kashihara's open problem number $30$, concerning the sequence $A001292$ of the OEIS, asking how many terms are powers of integers. We confirm his last conjecture up to the $100128$-th term and provide a…

On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. We prove that the number of prime terms in the sequence is uniformly bounded. When the…

For positive integers $n\geq k\geq t$, a collection $ \mathcal{B} $ of $k$-subsets of an $n$-set $ X $ is called a $t$-packing if every $t$-subset of $ X $ appears in at most one set in $\mathcal{B}$. In this paper, we give some upper and…

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x \ge R_n$, then the interval $(x/2,x]$ contains at least $n$ primes. We sharpen Laishram's theorem that $R_n < p_{3n}$ by proving that the maximum of…

We revisit the so-called "Three Squares Lemma" by Crochemore and Rytter [Algorithmica 1995] and, using arguments based on Lyndon words, derive a more general variant which considers three overlapping squares which do not necessarily share a…

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…

Let $q$ be an odd prime power. Combining the discussion of Varnavides and a recent theorem of Ellenberg and Gijswijt, we show that a subset $A\subset{\mathbb F}_q^n$ will contain many non-trivial three-term arithmetic progressions, whenever…

We improve the upper bound for the maximum possible number of stable matchings among $n$ jobs and $n$ applicants from $131072^n+O(1)$ to $3.55^n+O(1)$. To establish this bound, we state a novel formulation of a certain entropy bound that is…

Let $k$ be an even positive integer, $p$ be a prime and $m$ be a nonnegative integer. We find an upper bound for orders of zeros (at cusps) of a linear combination of classical Eisenstein series of weight $k$ and level $p^m$. As an…

Let $p$ be a prime number, and let $S$ be the numerical semigroup generated by the prime numbers not less than $p$. We compare the orders of magnitude of some invariants of $S$ with each other, e. g., the biggest atom $u$ of $S$ with $p$…

A primorial prime is a prime number of the form $p_n\# \pm 1$ where $p_n\#$ denotes the product of all primes less than or equal to $p_{n}$, the $n$-th prime. We show that the probability along the lines of Mertens' Theorem that either…

In this paper we study a sequence involving the prime numbers by deriving two asymptotic formulas and finding new upper and lower bounds, which improve the currently known estimates.

Generating primes is a fundamental problem in modern cryptography. Deterministic primality tests work well for special integers such as Mersenne or Proth primes, but these forms are quite restrictive. In this paper, we give a direct method…

From known effective bounds on the prime counting function of the form \[ |\pi(x)-\mathrm{Li}(x)| < a \;x \;(\ln x)^{b} \; \exp\left(-{c}\; \sqrt{\ln x}\right); \qquad (x \geq x_0); \] it is possible to establish exponentially tight…

Given $k, \ell \in {\bf N}^+$, let $x_{i,j}$ be, for $1 \le i \le k$ and $0 \le j \le \ell$, some fixed integers, and define, for every $n \in {\bf N}^+$, $s_n := \sum_{i=1}^k \prod_{j=0}^\ell x_{i,j}^{n^j}$. We prove that the following are…

Let $\mathbf H_2$ denote the set of even integers $n \not\equiv 1 \pmod 3$. We prove that when $H \ge X^{0.33}$, almost all integers $n \in \mathbf H_2$, $X < n \le X + H$ can be represented as the sum of a prime and the square of a prime.…