Question

If you take any positive integer and apply the following infinitely:

If the number is even, divide by two
If the number is odd, multiply by 3 and add 1

Will every integer eventually fall back down into the 4, 2, 1 loop, or is there a positive integer that never falls down into that loop?

Answer 1

This is known as the Collatz Conjecture, and it remains an unsolved problem in mathematics. Despite extensive computational evidence suggesting that the conjecture is true, a proof or counterexample has yet to be found.

The conjecture states that no matter what positive integer you start with, applying the "3n+1" rule (multiply by 3 and add 1 if n is odd) or "n/2" rule (divide by 2 if n is even) repeatedly will eventually lead to the sequence 4, 2, 1, and then it will loop endlessly: 4, 2, 1, 4, 2, 1, and so on.

While the conjecture has been checked for all starting values up to at least 10^20, no one has been able to prove that it holds true for all positive integers. It is possible that there exists a starting value that does not eventually fall into the 4, 2, 1 loop, but no such value has been found.