# Collatz series and noticed a pattern in the lengths

1. Sep 17, 2004

### Alkatran

I've been working with the Collatz series and noticed a pattern in the lengths. If you take the first occurence of a length, then find the first occurence of the next higher length after that, and continue... the steps will always be twice as large.

For example, from length 1 to length 2, the step is 1. From length 2 to length 3 the step is 2, and etc

I was just wondering what other patterns are known about it?

I'm current checking if the actual step in steps from length value to length value have some pattern...

2. Sep 17, 2004

### Alkatran

Ah, the steps increase linearly. This only stands for the first occurence of numbers, but the next value is always n higher, where n is the position of the length.

It's been shown that if you can show the value mod 2 = 0, that it MUST go to 1, right? (assuming that it is true, but not proven yet)
Since it divides to become a smaller number, and all numbers lower than the test number are proven... right?
(this wouldn't apply to numbers other than the initial value)

Last edited: Sep 17, 2004
3. Sep 17, 2004

### Alkatran

How would I prove:

n is odd and greater than 4
2^x > n, 2^(x-1) < n
x is greater than 2 (as a consequence)

(n*3 + 1) mod (2^x) = (n mod (2^x)) - 1

4. Sep 17, 2004

### Gokul43201

Staff Emeritus
This doesn't look right...

n=5, x=3

(15+1) mod 8 = 0 but (5 mod 8) - 1 = 4

But then I don't know anything about the Collatz Sequence (other than that Comp Geeks find it fun and it has something to do with numbers of the form 3n+1).

Last edited: Sep 17, 2004
5. Sep 18, 2004

### Alkatran

Change that -1 to a smaller than sign...

It was just one of those patterns you half see and need to check