Collatz Conjecture: The Collatz Conjecture, though simple to state, has baffled mathematicians since the 1930s. Start with any positive integer: if even, divide by 2; if odd, multiply by 3 and add 1. Repeat. The conjecture claims this process always reaches 1, regardless of the starting number. Verified for numbers up to 2⁶⁸, it shows chaotic, unpredictable behavior despite deterministic rules. No counterexample exists, yet no proof confirms universal convergence. The problem resists standard mathematical tools—number theory, dynamical systems, and probabilistic models offer partial insights but no resolution. Terence Tao made progress in 2019 by showing "almost all" orbits eventually decrease, but the full conjecture remains open. Its deceptive simplicity makes it a favorite among amateurs and professionals alike, symbolizing how elementary questions can conceal profound complexity.
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