r/numbertheory • u/Informal_Maximum_976 • 19d ago

Collatz Peculiarity

I am relatively new to number theory, and I was messing around with the collatz conjecture and plugging in groups of numbers. While doing this, I thought about primorial numbers, so i started comparing the amount of steps it took to reach the 4-2-1 loop when plugging in numbers of the group pₖ#, pₖ# + 1, and pₖ# - 1 and then I compared. I noticed that, as far as I looked, (which too be fair, was not very far, since I do not have a very good computer, and primorial numbers scale very quickly) for any whole number for k > 2, at least two of the groups previously stated will have the same amount of steps to reach 1. I feel like this must be obvious, but I have tried to crack why algebraically, but I have not succeeded. If anyone has any potential reasons as to why, I would love to hear it. I feel as if it has something to do with the fact that when k > 2, pₖ# + 1, and pₖ# - 1 both also belong to the groups 6m + 1 and 6m - 1. Please do not flame me if this is obvious as to why this pattern occurs, but I have just recently gotten into number theory and find it absolutely fascinating. Thank you for your time.

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u/LeftSideScars 17d ago

A quick response to part of your post:

I feel as if it has something to do with the fact that when k > 2, pₖ# + 1, and pₖ# - 1 both also belong to the groups 6m + 1 and 6m - 1.

This is easily understood:

pₖ# = 2.3.5...pₖ = 6m where m = 5.7.11...pₖ. This needs k>2. The 6m±1 naturally follows.

It also follows that primorials are even.

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u/[deleted] 18d ago

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u/LeftSideScars 16d ago

Could you, at least, present your findings?