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 main contribution of this paper is a proof that the function $\lceil{B^{c^n}}\rceil$ is also a prime-representing function, where $\lceil X\rceil$ denotes the ceiling or least integer function. Moreover, the first 10 primes in the sequence generated in the case $c=3$ are calculated. Lastly, the value of $B$ is approximated to the first $5500$ digits and is shown to begin with $1.2405547052\ldots$.
It is well known that the arithmetic nature of Mills' prime-representing constant is uncertain: we do not know if Mills' constant is a rational or irrational number. In the case of other prime-representing constants, irrationality can be proved, but it is not known whether these constants are algebraic or transcendental numbers. By using Liouville or Roth's theorems about approximation by rationals, we find a couple of prime-representing constants that can be proved to be transcendental numbers.
We prove that every positive integer $n$ which is not equal to $1$, $2$, $3$, $6$, $11$, $30$, $155$, or $247$ can be represented as a sum of a squarefree number and a prime not exceeding $\sqrt{n}$.
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 consider other sequences that can be generated using the same method.
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 $A= 1.3063778838\ldots .$ Here we give a first unconditional variant: $\lfloor A^{10^{10n}}\rfloor$ is prime, where $A=1.00536773279814724017\ldots$ can be computed to millions of digits. Similarly, $\lfloor A^{3^{13n}}\rfloor$ is prime, with $A=3.8249998073439146171615551375\ldots .$
Let c > 0.55. Every large n can be written in the form p +ab, where p is prime, a and b are significantly smaller than x^1/2 and ab is less than n^c. This strengthens a result of Heath-Brown, which has the requirement c>3/4. We introduce the idea of 'intersecting two sieves' as a tool in the proof of this result.
We prove that, for every modulus $\mathfrak{q}$, every class of the narrow ray class group $H_{\mathfrak{q}}(\mathbf{K})$ of an arbitrary number field $\mathbf{K}$ contains a product of three unramified prime ideals $\mathfrak{p}$ of degree one with $\mathfrak{N}\mathfrak{p}\le (t(\mathbf{K})\mathfrak{N}\mathfrak{q})^3$, where $t(\mathbf{K})$ is an explicit function of $\mathbf{K}$ described in the paper. To achieve this result, we first obtain a sharp explicit Brun-Titchmarsh Theorem for ray classes and then an equally explicit improved Brun-Titchmarsh Theorem for large subgroups of narrow ra…
In 2010, Matomäki investigated the set of $A>1$ such that the integer part of $ A^{c^k} $ is a prime number for every $k\in \mathbb{N}$, where $c\geq 2$ is any fixed real number. She proved that the set is uncountable, nowhere dense, and has Lebesgue measure $0$. In this article, we show that the set has Hausdorff dimension $1$.
Let $(c_k)_{k\in \mathbb{N}}$ be a sequence of positive integers. We investigate the set of $A>1$ such that the integer part of $A^{c_1\cdots c_k}$ is always a prime number for every positive integer $k$. Let $\mathcal{W}(c_k)$ be this set. The first goal of this article is to determine the topological structure of $\mathcal{W}(c_k)$. Under some conditions on $(c_k)_{k\in \mathbb{N}}$, we reveal that $\mathcal{W}(c_k)\cap [0,a]$ is homeomorphic to the Cantor middle third set for some $a$. The second goal is to propose an algebraically independent subset of $\mathcal{W}(c_k)$ if $c_k$ is rapidl…
We show that $$\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{1/2}}} (p_{n+1} - p_n) \ll x^{0.57+ε}$$ and $$\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{0.45}}} (p_{n+1} - p_n) \ll x^{0.63+ε},$$ where $p_n$ is the $n$th prime number. The proof combines Heath-Brown's recent work with Harman's sieve, improving and extending his results. We give applications of the results to prime-representing functions, binary digits of primes and approximation of reals by multiplicative functions.
Research Center in Mathematics and Applications (CIMA)（数学与应用研究中心）；Department of Mathematics, School of Sciences and Technology of the University of Évora（埃武拉大学科学与技术学院数学系）
本文研究了素数表示常数递归构造中出现的映射$\mathcal{T}(x)=\lfloor x floor(1+\{x\})$的有理动力学，定义了有理数的阶，并证明了有限阶有理数的自然密度为1，同时给出了特定分母下的显式分数家族。
Let $Λ(n)$ be the von Mangoldt function, $x$ real and $2\leq y \leq x$. This paper improves the estimate on the exponential sum over primes in short intervals \[ S_k(x,y;α) = \sum_{x< n \leq x+y} Λ(n) e\left( n^k α\right) \] when $k\geq 3$ for $α$ in the minor arcs. And then combined with the Hardy--Littlewood circle method, this enables us to investigate the Waring--Goldbach problem of representing a positive integer $n$ as the sum of $s$ $k$th powers of almost equal prime numbers, which improves the results in Wei and Wooley [12].
We describe an algorithm for arbitrary-precision computation of the elementary functions (exp, log, sin, atan, etc.) which, after a cheap precomputation, gives roughly a factor-two speedup over previous state-of-the-art algorithms at precision from a few thousand bits up to millions of bits. Following an idea of Sch{ö}nhage, we perform argument reduction using Diophantine combinations of logarithms of primes; our contribution is to use a large set of primes instead of a single pair, aided by a fast algorithm to solve the associated integer relation problem. We also list new, optimized Machin-l…
In this report we present an off-the-number-line representation of the positive integers by expressing each integer by its unique prime signature as a grid point of an infinite dimensional space indexed by the prime numbers, which we term the prime grid. In this space we consider a zigzag line, termed the number trail that starts at the origin (representing 1) and travels through every single grid point in the order of the increasing sequence of the natural numbers. Using the infinity norm we define an arithmetic function $L_\infty (N)$ tabulating the total length of the zigzag up to the integ…
