A Variation on Mills-Like Prime-Representing Functions

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 $1.2405547052\ldots$.