Yup, for the past two days, for the first time I thought about the problem! Yup, quickly it gets to the role of prime numbers or a product of primes and some n

Yup, for the past two days, for the first time I thought about the problem! Yup, quickly it gets to the role of prime numbers or a product of primes and some number of factors of 2. Heck, obviously can draw a directed graph with an arc from each odd number, that is, a product of primes except 2, and the result of the 3X + 1 operation. Then have an arc from each even number to that number with one less factor of 2. Then if could argue that given a number n, the nodes in the graph that can be reached from n are finite and if we don't enter cycles without a 1, then we are done! Alas, as one might expect from all the effort on this problem, a gap here is the argument of finiteness, and that quickly gets to how many primes there are or are not! And as we know, where all the primes are is a tough problem. Long ago in math, I decided not to spend effort on big questions about prime numbers! So, I gave up, decided to use easier problems to think about when going to sleep!