The conjecture can be summarized as follows. Take any positive integer n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called “Half Or Triple Plus One”, or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1.
The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after a Polish-American mathematician Stanislaw Ulam), Kakutani’s problem (after a Japanese-American mathematician Shizuo Kakutani), the Thwaites conjecture (after British Sir Bryan Thwaites), Hasse’s algorithm (after a German mathematician Helmut Hasse), or the Syracuse problem. The sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.
Paul Erdős said about the Collatz conjecture: “Mathematics may not be ready for such problems.”
Jeffrey Lagarias in 2010 claimed that based only on known information about this problem, “this is an extraordinarily difficult problem, completely out of reach of present day mathematics.”
Our main efforts in this area will be concentrated on testing and proving Collatz conjecture both theoretically and algorithmically.