Quote:
Originally Posted by dabler
I am concerned with the 7x±1 problem. My conjecture is that the orbit of arbitrary positive integer always leads to 1. I have computationally verified the conjecture for all numbers up to 10 ^{10}. I would appreciate hearing any thoughts on this. Thanks!

I hereby eat some crow. I read the problem as 7x+1, not 7x+/1 . The minus sign
got buried within the underline on my terminal. Its display is a bit fuzzy.
7x+/1 should converge because when we select +1 or 1 depending on 7x mod 4 we pick up an extra factor of 2. (7/8)^n as opposed to (7/4)^n behaves very differently.
I apologize profusely.
It is clear, however, that the pure 7x+1 iterations do NOT converge.