Are the Collatz and abc conjectures related?

TL;DR

This paper explores a potential connection between the Collatz and abc conjectures, demonstrating how assuming abc can improve bounds on certain Collatz sequences and involving Wieferich primes for finding large triples.

Contribution

It shows that the abc conjecture can be used to improve bounds on specific Collatz sequences and introduces a subset of abc-hits relevant to this analysis.

Findings

Assuming abc improves lower bounds for certain Collatz sequences

A small subset of abc-hits is useful for sharper bounds

Wieferich primes help find large triples in the subset

Abstract

The Collatz and $abc$ conjectures, both well known and thoroughly studied, appear to be largely unrelated at first sight. We show that assuming the $abc$ conjecture true is helpful to improve the lower bound of integers initiating a particular type of Collatz sequences, namely finite sequences of a given length where all terms but one are odd with the usual ``shortcut'' form. To obtain sharper bounds in this context, we are led to consider a small subset of the $abc$-hits. Then, it turns out that Collatz iterations as well as Wieferich primes may be used to find large triples in this subset.