The problem
Define \(f(n) = n/2\) for even \(n\), \(f(n) = 3n+1\) for odd \(n\). Starting from any positive integer and iterating \(f\): must the orbit reach the cycle \(1 \to 4 \to 2 \to 1\)?
The Collatz Conjecture
openHistory & significance
Named after Collatz, who circulated the iteration in 1937; popularised as the 3x+1 problem. Verified far beyond 10²⁰ without a hint of failure. Erdős: "Mathematics may not be ready for such problems." Tao (2019) proved almost all orbits get arbitrarily close to 1 in logarithmic density — celebrated, yet nowhere near enough.
Connected problems
Still open.
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