# On duplicate representations as $2^x + 3^y$ for nonnegative integers $x$ and $y$

## TL;DR

This paper proves that only five positive integers can be expressed as the sum of a nonnegative power of 2 and a nonnegative power of 3 in more than one way, confirming a conjecture from the OEIS.

## Contribution

The paper establishes the exact count of integers with multiple representations as sums of powers of 2 and 3, including the case with zero exponents, using elementary methods.

## Key findings

- Exactly five positive integers have multiple representations.
- The case with positive exponents is known from Bennett's theorem.
- Elementary methods suffice to prove the case with zero exponents.

## Abstract

We prove a conjecture posted in the Online Encyclopedia of Integer Sequences, namely that there are exactly five positive integers that can be written in more than one way as the sum of a nonnegative power of 2 and a nonnegative power of 3. The case for both powers being positive follows from a theorem of Bennett. We use elementary methods to prove the case where zero exponents are allowed.