The Collatz Conjecture

Lets take a positive integer.

Now keep applying these rules repeatedly.

Statement:- This process will eventually reach the number 1, regardless of which positive integer is chosen initially.

(that is: a_i is the value of f applied to n recursively i times; a_i = fⁱ(n)).

That smallest i such that a_i = 1 is called the total stopping time of n. The conjecture asserts that every n has a well-defined total stopping time. If, for some n, such an i doesn’t exist, we say that n has infinite total stopping time and the conjecture is false. If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found.

For example starting with 12 we get the sequence 12, 6, 3, 10, 5, 16, 8, 4, 2, 1.

The sequence of numbers involved is sometimes referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.

 “Mathematics may not be ready for such problems.”- Paul Erdős

Defining the Collatz function f(x) as follows:

If x is a positive integer, then f(x) is the next number after x in its Collatz sequence.

To extend this function to the real numbers, simply recall that (-1)^x = cos(πx). In fact, this gets us an extension to the complex numbers at the same time, and after some simplification we arrive at:

It is a holomorphic function and we can study the fractal that its iterates induce.

The fractal is located on the complex plane, and the horizontal line through the middle of the image is the real line. Black regions are regions in which the orbit of that number is bounded, while other colors indicate that the orbit of that number is unbounded (notice the large region of bounded numbers around z = 0). The big “spikes” that occur along the real line are, as we would expect, located at the integers (the image above is wide enough that you can see the spikes at z = -2, -1, 1, and 2).

The Collatz Conjecture is quite intimately connected with deep roots of nature and much more lies in the dark left to discovered about it which includes a PROOF.