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\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}
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\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