\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics,amsmath,amssymb} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsf} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.1in} \setlength{\textheight}{8.4in} \newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \begin{center} \vskip 1cm{\LARGE\bf A Variation on Mills-Like \\ \vskip .1in Prime-Representing Functions } \vskip 1cm \large L\'aszl\'o T\'oth\\ Rue des Tanneurs 7 \\ L-6790 Grevenmacher \\ Grand Duchy of Luxembourg \\ \href{mailto:uk.laszlo.toth@gmail.com}{\tt uk.laszlo.toth@gmail.com} \end{center} \vskip .2 in \begin{abstract} 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 $1.2405547052\ldots$. \end{abstract} \section{Introduction} Mills \cite{Mills47} showed in 1947 that there exists a constant $A$ such that $\lfloor{A^{3^n}}\rfloor$ is prime for all positive integers $n$. Kuipers \cite{Kuipers50} and Ansari \cite{Ansari51} generalized this result to all $\lfloor{A^{c^n}}\rfloor$ where $c\in\mathbb{R}, c\geq2.106$, i.e., there exist infinitely many $A$'s such that the above expression yields a prime for all positive integers $n$. Caldwell and Cheng \cite{CaldwellCheng05} calculated the minimum constant $A$ for the case $c=3$ up to the first $6850$ digits (\seqnum{A051021}), and found it to be approximately equal to $1.3063778838\ldots$. This process involved computing the first $10$ primes $b_i$ in the sequence generated by the function (\seqnum{A051254}), with $b_{10}$ having 6854 decimal digits. The main contribution of this paper is a proof that the function $\lceil{B^{c^n}}\rceil$ satisfies the same criteria, where $\lceil X\rceil$ denotes the ceiling function (the least integer greater than or equal to $X$). In other words, there exists a constant $B$ such that for all positive integers $n$, the expression $\lceil{B^{c^n}}\rceil$ yields a prime for $c\geq 3, c\in\mathbb{N}$. Moreover, the sequence of primes generated by such functions is monotonically increasing. Lastly, analogously to \cite{CaldwellCheng05} the case $c=3$ is studied in more detail and the value of $B$ is approximated up to the first $5500$ decimal digits by calculating the first $10$ primes $b_i$ of the sequence. In contrast to Mills' formula and given that here the floor function is replaced by a ceiling function, the process of generating the prime number sequence $P_0, P_1, P_2, \ldots$ involves taking the greatest prime smaller than $P_n^c$ at each step instead of smallest prime greater than $P_n^c$, in order to find $P_{n+1}$. As a consequence, the sequence of primes generated by $\lceil{B^{c^n}}\rceil$ is different from the one generated by $\lfloor{A^{c^n}}\rfloor$ for the same value of $c$ and the same starting prime (apart from the first element of course). \section{The prime-representing function} This paper begins with a proof of the case $c=3$ and will proceed to a generalization of the function to all $c\geq 3, c\in\mathbb{N}$. By using Ingham's result \cite{Ingham37} on the difference of consecutive primes: $$ p_{n+1} - p_n < Kp_n^{5/8}, $$ and analogously to Mills' reasoning \cite{Mills47}, we construct an infinite sequence of primes $P_0, P_1, P_2, \ldots$ such that $\forall n \in \mathbb{N} : (P_n-1)^3+1 < P_{n+1} < P_n^3$ using the following lemma. \begin{lemma}\label{bounds} $\forall N > K^8+1 \in \mathbb{N} : \exists p \in \mathbb{P} : (N-1)^3+1 K^8 +1) \\ & < N^3 - 2N^2 + N \\ & < N^3. \end{align*} Note that since $(N-1)^3 < p_{n+1}$, $(N-1)^3+1 < p_{n+1}$ since $(N-1)^3+1 = N(N^2-3N+3)$ is not prime. \end{proof} Given the above we can construct an infinite sequence of primes $P_0, P_1, P_2, \ldots$ such that for every positive integer $n$, we have: $(P_n-1)^3+1 < P_{n+1} < P_n^3$. We now define the following two functions: \begin{align*} \forall n \in \mathbb{Z^+}: u_n &= (P_n-1)^{3^{-n}}, \\ \forall n \in \mathbb{Z^+}: v_n &= P_n^{3^{-n}}. \end{align*} The following statements can immediately be deduced: \begin{itemize} \item $u_n < v_n$, \item $u_{n+1} = (P_{n+1}-1)^{3^{-n-1}} > \left((P_n-1)^3+1)-1\right)^{3^{-n-1}} = (P_n-1)^{3{-n}} = u_n$, \item $v_{n+1} = P_{n+1}^{3^{-n-1}} < (P_n^3)^{3^{-n-1}} = P_n^{3^{-n}} = v_n$. \end{itemize} It follows that $u_n$ forms a bounded and monotone increasing sequence. \begin{theorem} \label{theorem3n} There exists a positive real constant $B$ such that $\lceil{B^{3^n}}\rceil$ is a prime-representing function for all positive integers $n$. \end{theorem} \begin{proof} Since $u_n$ is bounded and strictly monotone, there exists a number $B$ such that $$ B := \lim_{n\rightarrow\infty}u_n. $$ From the above deduced properties of $u_n$ and $v_n$, we have \begin{alignat*}{2} u_n &< B &&< v_n, \\ (P_n-1)^{3^{-n}} &< B &&< P_n^{3^{-n}}, \\ P_{n}-1 &< B^{3^n} &&< P_n. \end{alignat*} \end{proof} \begin{theorem} There exists a positive real constant $B$ such that $\lceil{B^{c^n}}\rceil$ is a prime-representing function for $c\geq 3, c\in\mathbb{N}$ and all positive integers $n$. \end{theorem} \begin{proof} We can use the generalizations to Mills' function as shown by Kuipers \cite{Kuipers50} and Dudley \cite{Dudley69} in order to show that $\lceil{B^{c^n}}\rceil$ is also a prime-representing function for $c\geq 3, c\in\mathbb{N}$. This proof is short as it is essentially identical to the one presented above, with the following modifications. As shown by Kuipers \cite{Kuipers50} for Mills' function, we first define $a=3c-4, b=3c-1$. Therefore $a/b\geq 5/8$. This means that in Ingham's equation there exists a constant $K'$ such that $$ p_{n+1} - p_n < K'p_n^{a/b}. $$ Lemma \ref{bounds} can then be modified by taking $N>K'^b+1$, defining $p_n$ as the greatest prime smaller than $(N-1)^c$ and noticing that $ca+1 = b(c-1)$. Analogously to the proof in Lemma \ref{bounds}, we quickly obtain the bounds $(N-1)^c+1