Results on the Collatz Conjecture

Abstract

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] Given a starting value, we can create a sequence using the rule that if the previous number, x, is even, then the next number is [x/2], and if the previous number, x, is odd, then the next number is [(3x+1)/2]. The collatz conjecture is that any such sequence will eventually hit 1. We show that if such a sequence is bounded, then it will either always hit 1 or there will be a sequence with starting with a number of the form 3n + 1 which hits 3n + 1 again in less than 6n steps. [Formulae in brackets reformatted due to lack of available characters in character set used.]