I've been fiddling with the Collatz Conjecture off and on for years now. I'm convinced I found a pattern that I haven't been able to find mentioned anywhere. Granted, that could be because I lack the mathematical language needed to search for it.
First, I'm going to use an implicit even step after the odd step, as 3*odd + 1 always equals even. If you look at the path a number takes to its next lowest number, for example 5->8->4, visualize it by just looking at the even and odd steps like so: 5->10, you will see that other numbers follow a similar pattern:
9->10
13->10
17->10
What do these number have in common? They follow the pattern 5 + k(2^n) where n is the number of even steps (with the implicit even step, two in this case).
For another example, look at 7:
7->1110100
Seven even steps, so the next number will be 7 + 2^7 = 135:
135->1110100
I'd love to hear if this has been found and documented somewhere. If not, I have additional ramblings to share.
First, I'm going to use an implicit even step after the odd step, as 3*odd + 1 always equals even. If you look at the path a number takes to its next lowest number, for example 5->8->4, visualize it by just looking at the even and odd steps like so: 5->10, you will see that other numbers follow a similar pattern:
9->10
13->10
17->10
What do these number have in common? They follow the pattern 5 + k(2^n) where n is the number of even steps (with the implicit even step, two in this case).
For another example, look at 7:
7->1110100
Seven even steps, so the next number will be 7 + 2^7 = 135:
135->1110100
I'd love to hear if this has been found and documented somewhere. If not, I have additional ramblings to share.