Collatz conjecture

The Collatz conjecture claims that repeatedly halving even positive integers and replacing odd ones with three times the number plus one eventually reaches 1.

Collatz map: The function that sends an integer n to n/2 when n is even and to 3n+1 when n is odd. It defines the step repeated in the conjecture.
Lothar Collatz: A German mathematician who introduced the iteration underlying the conjecture in the 1930s. His name became attached to the problem.
Terence Tao: An Australian-American mathematician whose work spans analysis, number theory, and combinatorics. He proved that almost all Collatz orbits attain values far below their starting points, in a precise density sense.
Parity (mathematics): The property of an integer being even or odd. Parity selects which Collatz rule applies at each step.
Computational verification of the Collatz conjecture: Computer-assisted checking that the Collatz conjecture holds for every starting integer up to a stated bound. Finite verification establishes many cases but cannot settle the universal claim.
Collatz sequence: A sequence formed by iterating the Collatz map from a chosen positive integer. The conjecture asks whether every such sequence reaches 1.
Shizuo Kakutani: A Japanese-American mathematician associated with publicizing the Collatz problem in the United States. His circulation of the problem helped spread its reputation.
Jeffrey Lagarias: An American mathematician who has written extensively about the 3x+1 problem. His surveys and results have organized much of the problem’s mathematical literature.
Modular arithmetic: Arithmetic in which integers are considered equivalent when they have the same remainder modulo a fixed number. Congruence classes help describe possible sequences of odd and even steps.
Almost all (mathematics): A phrase indicating that exceptions form a negligible set under a specified notion of size or density. Tao’s theorem controls almost all starting values without proving the conjecture for every one.

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