The Collatz 3n+1 problem

Abstract:

A positive integer a0a0 determines recursively the sequencea0,a1,a2,…a0,a1,a2,… by the Collatz rules an+1=an/2an+1=an/2 for even nn and an+1=(3an+1)an+1=(3an+1) fornn odd.  Massive electronic calculation over many years has verified that for each`start' a0a0 examined there is an n≥0n≥0 with an=1an=1 and so an+3=1an+3=1, etc. Theauthor's experience in using modern computers to understand WW2 cryptology hasbeen used to find a new phenomenon in this context.