Powers of 3 with few nonzero bits and a conjecture of Erd\H{o}s

TL;DR

This paper uses elementary methods to classify powers of 3 and 2 that can be expressed as sums of a limited number of powers of the other base, providing insights related to Erdős's conjecture.

Contribution

It completely characterizes such powers using elementary techniques, offering a new perspective on exponential Diophantine equations and Erdős's conjecture.

Findings

All powers of 3 that are sums of at most 22 powers of 2.

All powers of 2 that are sums of at most 25 powers of 3.

Support for Erdős's conjecture regarding powers of 2 as sums of powers of 3.

Abstract

Using completely elementary methods, we find all powers of 3 that can be written as the sum of at most twenty-two distinct powers of 2, as well as all powers of 2 that can be written as the sum of at most twenty-five distinct powers of 3. The latter result is connected to a conjecture of Erd\H{o}s, namely, that 1, 4, and 256 are the only powers of 2 that can be written as a sum of distinct powers of 3. We present this work partly as a reminder that for certain exponential Diophantine equations, elementary techniques based on congruences can yield results that would be difficult or impossible to obtain with more advanced techniques involving, for example, linear forms in logarithms.