Sample number space of positive integer x’s between 1 and any arbitrary (positive) integer) n, and find Max(3x+1) for each iteration for all trials for all values of x between 1 and n.
.
For any arbitrary large b, there is always an equal or larger Max(3(x+b)+1) between 1 and (x+b) compared to that found between 1 and x.
Therefore, As (x+b) -> infinity, (3(x+b)+1) -> infinity.
COROLLARY
If one samples the same positive integer number space between 1 and any arbitrary n, one finds Max(Stopping I) for each iteration for all trials involving 1 to n.
For any arbitrary large b, there is always an equal or larger Max(Stopping I) between 1 and (x+b) compared to 1 and x.
This tells us that as (x+b) -> infinity, (Stopping I) -> infinity
Cf proof: For any positive values of x and y, (x+y) > x. Further, as y -> infinity, (x+y) -> infinity.
I hope this helps.
James Lyons-Weiler
Allison Park, PA
8/2/2021
One comment