Exponential prime sequences

TL;DR

This paper proves the existence of infinite exponential prime sequences of specific forms under Cramer's conjecture, focusing on sequences with minimal growth rates and particular parameter constraints.

Contribution

It establishes the existence of four specific infinite exponential prime sequences with minimal growth parameters, assuming Cramer's conjecture on prime gaps.

Findings

Existence of four explicit exponential prime sequences.

Sequences have minimal possible growth rates under given constraints.

Assumption of Cramer's conjecture is crucial for proofs.

Abstract

Infinite exponential sequences of distinct prime numbers of the form $\lfloor a c^{n^d}+b\rfloor$, $n\geq 0$, are proved to exist for well chosen real constants $a>0$, $b$, $c>1$, $d>1$, assuming Cramer's conjecture on prime gaps. There is an infinity of such prime sequences. Sequences having the least possible growth rate are of particular interest. This work's focus is on prime sequences with $a=1$, $b \in \{0,1\}$, that have the smallest possible constant $c$ given $d>1$, and sequences with the smallest possible $d$, given $c=2$. In particular, we prove the existence of the four infinite exponential prime sequences $u_0(n)=\lfloor c_0^{n\sqrt{n}}\rfloor$, $n\geq 1$, with $c_0=2.0073340803...$, $u_1(n)=1+\lfloor c_1^{n\sqrt{n}}\rfloor$, $n\geq 0$, with $c_1=2.2679962677...$, $v_0(n)=\lfloor 2^{n^{d_0}}\rfloor$, $n\geq 1$, with $d_0=1.5039285240...$, and $v_1(n)=1+\lfloor 2^{n^{d_1}}\rfloor$, $n\geq 0$, with $d_1=1.7355149500...$.