Related papers: A Variation on Mills-Like Prime-Representing Funct…

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=…

In 1947 Mills proved that there exists a constant $A$ such that $\lfloor A^{3^n} \rfloor$ is a prime for every positive integer $n$. Determining $A$ requires determining an effective Hoheisel type result on the primes in short intervals -…

Let $ \lfloor x \rfloor $ denote the integer part of $ x $. In 1947, Mills constructed a real number $ \xi > 1 $ such that $\lfloor \xi^{3^k} \rfloor$ is always a prime number for every positive integer $k$. We define Mills' constant as the…

Let $[\, \cdot\,]$ be the floor function. In this paper, we prove that when $1<c<\frac{16559}{15276}$, then every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p^c_1]+[p^c_2]+[p^c_3]\,,…

We present a simple, closed formula which gives all the primes in order. It is a simple product of integer floor and ceiling functions.

Let $\lfloor x\rfloor$ denote the integer part of $x$. For every sequence $(C_k)_{k\ge 1}$ of positive integers, we define $\xi(C_k)$ as the smallest real number $\xi>1$ such that $\lfloor \xi^{C_k} \rfloor$ is a prime number for every…

Let $[\, \cdot\,]$ be the floor function. In this paper, we show that when $1<c<37/36$, then every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p^c_1]+[p^c_2]+[p^c_3]\,, \end{equation*} where…

The analysis of regularities and randomness in the distribution of prime numbers remains at the research frontiers for many generations of mathematicians from different groups and topical fields. In 2019 D. Fridman et al. (Am. Math. Mon.…

For n=1,2,3,... define S(n) as the smallest integer m>1 such that those 2k(k-1) mod m for k=1,...,n are pairwise distinct; we show that S(n) is the least prime greater than 2n-2 and hence the value set of the function S(n) is exactly the…

Let $${\mathbb P}^c=(\lfloor p^c\rfloor)_{p\in{\mathbb P}} \qquad (c>1,\ c\not\in {\mathbb N}), $$ where ${\mathbb P}$ is the set of prime numbers, and $\lfloor\cdot\rfloor$ is the floor function. We show that for every such $c$ there are…

Wright proved that there exists a number $c$ such that if $g_0 = c$ and $g_{n+1} = 2^{g_n}$, then $\lfloor g_n \rfloor$ is prime for all $n > 0$. Wright gave $c = 1.9287800$ as an example. This value of $c$ produces three primes, $\lfloor…

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.

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…

Let $[\, \cdot\,]$ be the floor function. In this paper we show that every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p_1\log p_1]+[p_2\log p_2]+[p_3\log p_3], \end{equation*} where $p_1,\,…

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.

Let $[\, \cdot\,]$ be the floor function. In this paper, we show that when $1<c<\frac{82}{79}$, then every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p^c]+[m^c]\,, \end{equation*} where $p$…

In this paper we introduce a new diophantine equation with prime numbers. Let $[\, \cdot\,]$ be the floor function. We prove that when $1<c<\frac{23}{21}$ and $\theta>1$ is a fixed, then every sufficiently large positive integer $N$ can be…

The divisor function $\sigma(n)$ denotes the sum of the divisors of the positive integer $n$. For a prime $p$ and $m \in \mathbb{N}$, the $p$-adic valuation of $m$ is the highest power of $p$ which divides $m$. Formulas for…

We present a constant and a recursive relation to define a sequence $f_n$ such that the floor of $f_n$ is the $n$th prime. Therefore, this constant generates the complete sequence of primes. We also show this constant is irrational and…

Hilberdink showed that there exists a constant $c_0>2$, such that there exists a continuous prim system satisfying $N(x)=c(x-1)+1$ if and only if $c\leq c_0$. Here we determine $c_0$ numerically to be $1.25479\cdot 10^{19}\pm2\cdot…