In most kinds of math, after using the knowledge we have, we either find or construct problems that prove that we’re wrong, or that our knowledge is incomplete. Some of them are very long and complicated, with very intricate formulas and ideas within them, along with similarly complex proofs to assure the correct solution. But that, right there, is the tricky part. Proofs.
To construct a proof is a delicate task that requires time, skill, and lots and lots of scrutiny. Some may seem over the top for what seems like a simple problem to solve. But, above all else, there are problems we don’t have the proof to. Or even anything resembling a solution. Welcome to the world of unsolved problems that can’t yet be proven, nor disproven. These have a special name: conjectures.
Today we will look at one of the most famous conjectures ever proposed, the Collatz Conjecture. It was proposed in 1937 by a German mathematician named Lothar Collatz, and the premise is very simple. Pick any counting number, as big as you want. If the number is even, divide by two. If it is odd, multiply by three and add one. Repeat. The conjecture states that every time, no matter what you choose, based on these parameters, you will always end up in this same loop. Four, because it’s even, will divide and become two. Once again, you divide and it becomes one. Now that the number we have is odd, we multiply by three and add one, so we end up back at four.
That may seem very simple, but we just can’t prove that it’s true. Using lots of computing, we have checked every consecutive number all the way up to 2^71, about 2.36 sextillion numbers. Even after all of that, we still cannot find any contradiction. In one case, researchers even checked the astronomically big number (2^100000)-1, which has a grand total of 30,103 digits, using a special set of bit manipulation algorithms. But even if we kept checking, we still wouldn’t be conclusive.
A proof is needed to decide whether this conjecture is true or false. But why do we need a proof? Can’t we see that all the numbers we check follow the pattern? While such ways of thinking may apply to other parts of the world, this doesn’t apply to math. In this system, if you pick any number randomly, you won’t really be able to predict how long it will take to get to the 4-2-1 loop. It is basically random. The number 26, for example, takes 10 steps. But 27, on the other hand, takes 111 steps to get to 1. This also shows that knowing any number, n, doesn’t affect your knowledge of n plus one, as that result may be completely different.
To prove this conjecture false, you have to find at least one of two things. The first is a number that shoots to infinity, and never comes down, so it will never hit a power of two. The second is a different loop, where a number appears twice in a sequence. But even after 80 years of trial and error, we still haven’t found a solution to this age-old problem. Maybe you will be the one to finally solve this mystery.