mersenneforum.org A new problem similar to the Collatz problem
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 2018-07-27, 21:03 #1 dabler     "David Barina" Jul 2016 Brno 4010 Posts A new problem similar to the Collatz problem 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 1010. I would appreciate hearing any thoughts on this. Thanks!
2018-07-28, 14:03   #2
R.D. Silverman

Nov 2003

22×5×373 Posts

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 1010. 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.

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