Revisiting the number of simple $K_4$-groups

TL;DR

This paper explores the difficulty of proving the infinitude of simple $K_4$-groups by analyzing related Diophantine equations, highlighting the problem's complexity beyond current conjectures.

Contribution

It demonstrates the challenges in establishing the infinitude of simple $K_4$-groups through Diophantine equations, indicating the problem's depth beyond existing conjectures.

Findings

Difficulty in proving infinitude of simple $K_4$-groups

Connection to complex Diophantine equations

Current limitations beyond Dickson's conjecture

Abstract

In this paper, by solving Diophantine equations involving simple $K_4$-groups, we will try to point out that it is not easy to prove the infinitude of simple $K_4$-groups. This problem goes far beyond what is known about Dickson's conjecture at present.