Pick any positive whole number. If it’s odd, multiply it by 3 and add 1. If it’s even, divide it by two. Apply these same rules to the new number you get and keep doing so. For example, I’ll choose the number 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1 → 4 → 2 → 1…
Eventually you will reach a dead end, a continuous loop of 421; or so the Collatz conjecture claims. This seemingly obvious claim has stumped mathematicians for almost 90 years, since it was first proposed in 1938 by Lothar Collatz. This conjecture has become so infamous in the mathematics community that mathematicians will often warn their students from attempting to prove it, which may explain the allure some people feel towards it. In fact during the cold war, the American government was halfconvinced that the whole problem was a Soviet scheme that was hatched to distract American mathematicians.
So if it hasn’t been proven how did I know that no matter what number you picked would end up in that loop? Computers have confirmed that this statement is true for the first 2⁶⁸ natural numbers, so unless you picked a number north of 300 quintillion you would reach 421, and if you did, good luck calculating that any time soon.
If computers haven’t found a counterexample proving it false can’t we assume that the conjecture is true? Unfortunately no, in the grand scheme of all natural numbers, 2⁶⁸ is insignificant. However it does raise an interesting question; what would a counterexample even look like? As a matter of fact there are 2 possible cases:

A number exists that increases boundlessly. We call this divergence.

There is another, or perhaps even multiple other loops that would prevent the sequence from reaching 421. Thanks to computer calculations, we know that this loop would have to be at least 186,000,000,000 numbers long
I find the second case particularly interesting, because we are yet to find any evidence that another loop couldn’t exist. In fact, if you apply the same two rules of the collatz conjecture, but on negative integers this time; there are three separate loops all starting at low values. Most of our mathematical resources are focused on proving the conjecture. But perhaps the reason why we have yet to reach any conclusive results is because almost no one is trying to disprove it. There’s also the possibility that we will never know, as Gödel’s incompleteness theorem tells us that there are true statements in mathematics which can never be proven.
But what is this conjecture useful for? As anticlimactic as it may sound there are actually only a few real world applications of this problem that we know of at this time. One application is benchmarking computers by observing how many integers they can calculate the sequence for. This begs the question of what can even be gained by proving this conjecture. Like with many math problems, new applications may surface after we have proved it. Additionally, we might discover new methods for solving problems, which could then be applied in other areas of mathematics.
Sharma, J. (2023, August 22). Looping and divergence in the Collatz conjecture. NHSJS. https://nhsjs.com/2022/loopinganddivergenceinthecollatzconjecture/