Unconditional Prime-representing Functions, Following Mills

Abstract

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 $A=3.8249998073439146171615551375\ldots .$