Problem Solving, Part 1

About 2 weeks ago, I began looking for a problem to solve, or at least attempt, for my blog. I hit a quick roadblock when I realized that I had no idea where to start looking. And so, I decided to attempt the first problem I found on Danny's problem solving wiki. The problem, in full, is as follows:
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Suppose n stands for some positive whole number. Define a function f(n) that associates n with another positive whole number, as follows:

• If n is even, then f(n) is n/2
• If n is odd, then f(n) is 3n+1

Since f(n) takes any positive whole number as input, and produces a positive whole number as output, it makes sense to talk about f(f(n)) or f(f(f(n))) --- composing f with itself several times. Let's define F(m,n) as f(f(...f(n)...), where f is composed with itself m times. Is the following statement true or false?
For each natural number n there exists a natural number m such that F(m,n) = 1.
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This is a problem that was very briefly mentioned in my calculus textbook. The main thing I remember is that, at my time of writing, it still is unsolved. Nonetheless, I will attempt it, hoping that I can at least make some progress in this deceptively simple problem.

The first step in solving any problem is to understand it. The basic idea behind this problem is that, if I continually apply f(n) to any positive natural number n, the number 1 will be reached in a finite amount of applications. The problem asks to prove or disprove this statement. Therefore, the problem can be solved by either:

Proving that there for every natural numbers n, there exists an m such that F(m,n) = 1

or

Proving that there exists some specific n such that for any m F(m,n) != 0

To convince myself that this function even works, I wrote a simple Python function that took the number from 1-5000 and plugged them into function, repeating until 1 was reached. (It turns out, of course, that it always was.  The number 3711 has the largest value for m, 237, but it still reached 1.)

My intuition would be that the statement is true. If the statement were false, then some computer would most likely have found the natural number and got caught in an infinite loop. After knowing this number, disproving the statement would be fairly easy, as one of the following would have to be true:

- There are infinitely many natural numbers n such that F(m,n) never reaches 0

or

- F(m,n) enters some sort of cycle that does not include 1.


So, my focus will be on proving that the statement is true. But first, I will try to find different forms of the statement that may be easier to work with.