Prime Holdout Problems

TL;DR

This paper introduces prime holdout problems related to the Collatz conjecture, providing proofs of convergence for certain cases and conjecturing about divergence, with implications for understanding the conjecture.

Contribution

It defines prime holdout problems, proves convergence for finite and infinite cases, and conjectures about divergence, linking to the Collatz conjecture.

Findings

Proved convergence for finite holdout problems

Proved convergence for infinite holdout problems

Conjectured that finite holdout problems cannot diverge

Abstract

This paper introduces prime holdout problems, a problem class related to the Collatz conjecture. After applying a linear function, instead of removing a finite set of prime factors, a holdout problem specifies a set of primes to be retained. A proof that all positive integers converge to 1 is given for both a finite and an infinite holdout problem. It is conjectured that finite holdout problems cannot diverge for any starting value, which has implications for divergent sequences in the Collatz conjecture.