I saw a thread earlier that mentioned Collatz, and as I have been working on that for some time now, I feel that I can share my work that I feel will be crucial towards any conceivable proof of Collatz. I lost my original notes since I've moved residences, however, this is an anecdote important enough that I've committed to memory.

There are infinitely many solutions to $4^{k} = 3n + 1$ such that k is a positive integer, and n is a positive integer (n is also odd by definition). For each value of k, there exists a value n that satisfies the equation. Presuming Collatz to be correct would imply that any value n that is a natural counting number will eventually reach a power of 4, and since 4 is a power of 2, the sequence will collapse to the trivial cycle.