Collatz

The Collatz problem, also known as the 3n+1 problem or the hailstone problem, concerns sequences formed by repeatedly applying a simple rule to a positive integer n: if n is even, replace it with n/2; if n is odd, replace it with 3n+1. The central conjecture is that every starting value eventually reaches 1, after which the sequence cycles through 1-4-2-1.

The problem was introduced by Lothar Collatz in 1937 and has since been widely studied. It is also known by other names, including the Syracuse problem, reflecting its independent appearance in different places.

Despite extensive numerical evidence, the conjecture remains unproven for all positive integers. It is known that the only nontrivial cycle observed in positive integers up to very large limits is 1-4-2-1; no proof has shown that no other cycles exist. The behavior of the sequences can be highly irregular, with values rising far above the starting number before eventually declining in many cases.

The Collatz function is simple to state but difficult to analyze. Sequences can rise to very large values before descending, and the problem has spurred various partial results on stopping times and average behavior. Computers have verified the conjecture for starting values up to at least 2^68.

In study and generalizations, researchers consider variants such as alternate formulas for odd terms or focus on reduced forms like Syracuse sequences. The Collatz problem remains a focal point of elementary number theory due to its deceptively simple rules and unresolved status.