The powers of 2 and 3 getting further away bit is not correct as you mention but it isn't actually all that important. My Github page is a WIP so bear with me..

The powers of 2 and 3 getting further away bit is not correct as you mention but it isn't actually all that important. My Github page is a WIP so bear with me.. If a cycle does exist for some large-ish cycle length, let’s say 100 odd numbers, then it means that (3+1/x1)(3+1/x2)…(3+1/x100) exactly equals some power of 2. Since (3+1/xn) rapidly approaches 3.0, for any large x1, the product of (3+1/x1)(3+1/x2) etc is going to be pretty close to a power of 3, but it needs to be much larger than that to equal a power of 2, since powers of 2 and 3 are always somewhat far apart. That means that x1, x2 etc must be relatively small so that the product of the (3+1/x) can actually be substantially larger than a power of 3. That puts a hard floor on the possible cycle values, and that floor actually trends smaller as the cycle gets larger since the distance between a power of 3 and the nearest power of 2 also trends larger.