Paul Erdős said about the Collatz conjecture: "Mathematics may not be ready for such problems." He also offered US$500 for its solution. The sequence of numbers involved is sometimes referred to as the hailstone sequence, hailstone numbers or hailstone numerals (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers. It is also known as the 3 n + 1 problem, the 3 n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem. It is named after mathematician Lothar Collatz, who introduced the idea in 1937, two years after receiving his doctorate. The conjecture is that these sequences always reach 1, no matter which positive integer is chosen to start the sequence. If the previous term is odd, the next term is 3 times the previous term plus 1. It concerns sequences of integers in which each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. The Collatz conjecture is one of the most famous unsolved problems in mathematics.
The Collatz conjecture states that all paths eventually lead to 1.
Directed graph showing the orbits of small numbers under the Collatz map, skipping even numbers.
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