Prime Holdout Problems

Josiah A. Yoder, Max Milkert, Alexander Ruchti

Abstract:

The Collatz conjecture, otherwise known as the 3n + 1 problem, is simple to state, yet has gone unsolved for over
70 years. Many researchers have attempted to reason about the conjecture by looking at generalizations, hoping to
have results that offer a solution to the specific problem. This paper takes a different approach, introducing a class of
problems opposite the Collatz conjecture that we refer to as holdout problems. The difference is that, after applying
a linear function, instead of dividing by a finite set of prime factors, a holdout problem specifies a set of primes to
be retained. A proof that all positive integer starting values converge to 1 is given for an example of a finite and
an infinite holdout problem. It is conjectured that finite holdout problems cannot diverge on any input, which has
implications for divergent sequences in the Collatz conjecture.