Gerhard Opfer has posted a paper that claims to resolve the famous Collatz conjecture.

Start with a positive number n and repeatedly apply these simple rules:

If n = 1, stop. If n is even, divide n by 2. If n is odd, multiply n by 3 and add 1.

In 1937, Lothar Collatz asked whether this procedure always stops for every positive starting value of n. If Gerhard Opfer is correct, we can finally say that indeed it always stops.

Update: It appears there’s a flaw in the proof. See discussion here. Perhaps the gap can be filled in, or perhaps an idea in the paper can be of use somewhere else.

Update (September 10. 2019): As of this date, the full Collatz conjecture remains unsolved, but Terence Tao has just posted a paper chipping away at the problem.

Related post: Easy to guess, hard to prove