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.

翻译：本文介绍了一类与Collatz猜想相关的问题——素数保留问题。不同于在应用线性函数后移除有限素数因子集，保留问题指定了一组需保留的素数。针对有限保留问题和无限保留问题，本文分别证明了所有正整数均收敛至1。我们推测有限保留问题对任何初始值均不会发散，该结论对Collatz猜想中的发散序列研究具有重要启示。