Collatz Conjecture
The 3n+1 problem — every orbit reaches 1 (unproven)
About: The Collatz conjecture (Lothar Collatz, 1937): take any positive integer. If even, divide by 2; if odd, multiply by 3 and add 1. Repeat. Conjecture: every starting number eventually reaches 1. Verified computationally for all n < 2⁶⁸, yet completely unproven — Paul Erdős remarked "Mathematics is not yet ready for such problems." The sequence for n=27 takes 111 steps and peaks at 9,232. Terence Tao (2019) proved almost all orbits eventually reach a value below any function growing to infinity — the closest result yet to a full proof.