A superpowered Euclidean prime generator

TL;DR

This paper discusses a variant of Euclid's prime generator that identifies the next prime after a product of the smallest primes using a specific divisor of a large exponential expression, exploring its properties and related methods.

Contribution

It introduces a novel prime generation method based on divisors of large exponential expressions and compares it with related prime-generating techniques.

Findings

The $(k+1)$-th prime can be obtained from the least divisor of $n^{n^n}-1$ where $n$ is the product of the first $k$ primes.

The method provides a new perspective on prime generation using exponential expressions.

Discussion of related prime-generating variants and their properties.

Abstract

When $k>1$ and $n$ is the product of the smallest $k$ primes, the $(k+1)$-st smallest prime is the least divisor exceeding $1$ of $n^{n^n}-1$. This variant of Euclid's prime generator is discussed with some of its cousins.