What would be the implications of (dis)proving the Collatz conjecture? The Collatz conjecture is one of the most famous unsolved problems in mathematics.  It essentially states that, for any positive integer, if you repeatedly apply the function 3x + 1 to it, factoring out any factors of 2 you run into along the way, you'll eventually reach a number that's a power of two.
It's a very interesting theoretical question, but if we ever found a proof (or a disproof), what would that do?  For instance, it's known that proving P = NP would have a huge effect on computability, and would render some cryptographic systems instantly obsolete that are currently thought to be secure due to depending on an NP-complete problem.  Would a (dis)proof of Collatz have any similar impacts on the world?
 A: Did anybody notice that the question's statement of the conjecture is incorrect? For example, starting with the integer $x_0 = 7$ and repeatedly applying $x_{n+1} = 3x_n+1$, you  never reach a power of 2 (because mod 16, you repeatedly get 7,6,3,2,7,6,3,2,... . The Collatz sequence for 7 terminates after only 5 steps of $3n+1$ (plus the requisite divisions by 2) but the addition of 1 and division by 2 do not commute with each other so the problem's description of the sequence is erroneous.
At any rate, if the cycle part of the conjecture is ultimately solved by the techniques of Simons, consisting of applying upper and lower bounds on the number of odd and even entries in an $m$-cycle, and noting that numerical results force the smallest value in such a sequence to be large so that the bounds contradict each other, then it seems that nothing new would be learned.
But it is more likely that a proof (especially the proof that there is no non-cyclic unbounded sequence) would involve some new technique or concept, in which case some important results might fall out.  As an example, Wiles' work on Fermat's last conjecture led to significant advances in the theory of modular forms, which might not have been found without that motivation.
