In my previous post, I spoke of the density of numbers which pass through a given number under iteration of the Collatz function. I wanted to clarify what I meant by that.
Let C(x, n) be the set of all numbers y which are less than n for which repeated iteration of y under the Collatz function yields x. For instance, if x = 5, then C(x, 12) would include 10 (since f(10) = 5), and would also include 3 (since f(f(3)) = f(10) = 5). However, it would not include 20, since 20 > 12.
Let d(x) (for density) be defined as the limit of the size of C(x, n)/n as n goes to infinity.
I have a new conjecture, and I'm working on a proof for it.
Conjecture: if x is equal to 0, 2, 3, or 4 mod 6, then d(x) = 0.
if x is equal to 1 or 5 mod 6, then d(x) > 0.
If I can prove this, then it shows that if there exist counterexamples to the Collatz conjecture, then they have a non-zero density. This is because it is possible to show that any even number will iterate to an odd number, and any odd number will iterate to a number which is 1 mod 6 or 5 mod 6. If x is a counterexample, then it is possible to generate y which is also a counterexample such that y = 1 or 5 mod 6. If there exists such a y, then every element of C(y, n) is a counterexample, and so the density of counterexamples would be greater than 0.
However, there is some evidence that counterexamples must be non-dense. These statements together could form a contradiction, thus proving that the Collatz conjecture is true.
No comments:
Post a Comment