The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 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. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that these sequences always reach 1, no matter which positive integer is chosen to start the sequence. The conjecture has been shown to hold for all positive integers up to 2.95×1020, but no general proof has been found.
The x axis represents starting number, the y axis represents the highest number reached during the chain to 1. This plot shows a restricted y axis: some x values produce intermediates as high as 2.7×107 (for x = 9663)
The number of iterations it takes to get to one for the first 100 million numbers.
A Collatz fractal centered at the origin, with real parts from -5 to 5.
Julia set of the exponential interpolation.
Lothar Collatz was a German mathematician, born in Arnsberg, Westphalia.
Lothar Collatz (Photo courtesy MFO)
Lothar Collatz in 1984
