Powers of 3 with few nonzero bits

Abstract:
Number theory is known as a field in which simple questions can sometimes only be answered using very advanced techniques. I’ll present an instance of a simple problem — going back in one form to a 14th century musicologist, and related to a 20th century conjecture of Erdős — whose solution seems unattainable with the high-powered techniques experts would first think of, but which can in fact be solved using only elementary arguments. The only difficulty involves*finding* the elementary solution, and carrying out the necessary computations.