In 1937, Lothar Collatz proposed the following conjecture: Start with a positive integer n, then repeatedly iterate the following: If n is even, divide it by 2; if n is odd, compute 3*n+1. Collatz conjectured that for every starting value n, the result will invariably return to 1.<\/p>\n
The Collatz conjecture has been studied by thousands of mathematicians and computer scientists. Portuguese mathematician Tomas Oliveira e Silva has verified the conjecture for all integers up to 5.76 x 10^18. But no proof has yet been found. Well-known mathematician Paul Erdos once characterized the Collatz conjecture as “Mathematics is not yet ready for such problems”. He offered a $500 reward for its solution.<\/p>\n
Now German mathematician Gerhard Opfer (who coincidentally was a student of Collatz) has announced that he has proven the conjecture. A draft of his paper, which has been submitted to a journal for review, is available here<\/a>.<\/p>\n Some additional information is available from this New Scientist article<\/a>, from which some of the information above was excerpted. Additional information is given in an article by Jeffrey Lagarias, which is available here<\/a>.<\/p>\n