Representing the integers with powers of 2 and 3

Abstract

The Collatz sequence (n 0,n 1,...) is defined by n k+1 = 3n k + 1 or n k /2 depending on n k being odd or even, respectively. The Collatz conjecture (one of the most challenging open problems in Number Theory) states then that n k = 1 for some k depending on n 0. This conjecture can be reformulated in a variety of ways, some of them seemingly more amenable to the methods of discrete mathematics. In this paper, we derive one such equivalent formulation involving exponential Diophantine equations. It follows that if the Collatz conjecture is true, then any number can be represented as sums of positive powers of 2 and negative powers of 3.