The Collatz conjecture
Start with any positive integer . If is even, replace it by , and if is odd, replace it by . Repeat this forever. Prove or disprove that every starting value eventually reaches 1.
History and references
Still open as of 2026.
Lothar Collatz introduced the iteration in 1937, and it has become the standard example of a problem anyone can state and nobody can prove. Computers have checked every starting value up to about without finding a counterexample, which would be either a cycle other than 4, 2, 1 or an orbit that grows without bound. A heuristic argument says that along an orbit each odd term is on average about three quarters of the previous odd term, so orbits should shrink. Turning this into a proof fails because the parity pattern of an orbit is not actually random. The strongest general result is Terence Tao's 2019 theorem that almost all orbits, in the sense of logarithmic density, fall below any function of the starting value that tends to infinity, however slowly.
References
Solutions 0
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