15.9.16

Collatz Exit

Definition: An exit point in relation to the Collatz conjecture is the last odd integer reached before reaching a power of 2.

Example: Let s = 10 then the Collatz exit point is 5 since 10/2 = 5 and 3*5 + 1 = 16, which is a power of 2.

The first few Collatz exit points are 5,21,85,341,1365,2731,5461,21845,87381

The sequence of Collatz exit points does not exist in the OEIS database.

Claim: If 2r - 1 is divisible by 3, then (2r - 1)/3 is a Collatz exit point.

Proof: A Collatz exit point is an odd integer x such that 3x + 1 is equal to a power of 2.
Let k = 2r - 1. Since k is divisible by 3 then k/3 is an integer i = k/3 therefore 3i = k and since k = 2r - 1 then 3i = 2r - 1 so 3i + 1 = 2r  .'.

Conjecture: If r is an even integer greater than 2, then 2r - 1 is divisible by 3

Definition: An exit path in relation to the Collatz conjecture is the last few integers reached in a Collatz trajectory starting from the Collatz exit point.

Example: Let s  = 10 then its Collatz exit path is 5,16,8,4,2,1

Note: In my previous blog entry I used the term "Collatz path" but I decided that "Collatz exit path" is a more descriptive term.

Definition: A Collatz trap is an odd integer n such that the last few powers of (n+1)/2 correspond to a Collatz exit path.

Example: Let n = 27, the last few powers of 14 mod 27 are 5, 16, 8, 4, 2, 1, which correspond to the Collatz exit path of 10.

The first few Collatz traps are: 27, 107, 427, 1707, 6827, 27307, 109227, 436907.

It appears that the sequence of Collatz traps already exists in the OEIS database but not in connection with the Collatz conjecture. I find this fascinating.

With these definitions in mind, I refine my previous algorithm for finding a Collatz trap from a Collatz exit path as follows:

Claim: Given a Collatz exit path [m, 2r, 2r-1, 2r-2, ... , 1] then a Collatz trap n is equal to:

n = (2r + 1) + ((2r - 1) - m)

Claim: Given a Collatz exit point m, then a Collatz trap is equal to 5*m + 2

Claim: For each unique Collatz exit point m, there exists a unique Collatz trap.