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

2019/07/07 by Douglas E. Iannucci, Iannucci, Douglas Edward
Computer Science · Mathematics · #11D61 #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1907.03347

openalex publication_date 2019/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Powers of 3 with few nonzero bits and a conjecture of Erdős
• Partitions of the set of nonnegative integers with the same\n representation functions
• Representations of integers as quotients of sums of distinct powers of three
• Representation by sums of unlike powers
• Sums of two squares and a power