Powers of 3 with few nonzero bits and a conjecture of Erdős

2021/05/13 by Vassil S. Dimitrov, Everett W. Howe, Dimitrov, Vassil S. +1 · 1 citation
Computer Science · Mathematics · #11D61 (Primary) 11A63 #11D72 #11D79 (Secondary) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Commutative Algebra and Its Applications #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2105.06440

openalex publication_date 2021/05/13 · openalex created_date 2023/05/18 · openalex updated_date 2026/07/28

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 Erdős, 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.

Citations

• Modular Elliptic Curves and Fermat's Last Theorem

Cited by

• Purely exponential Diophantine equations with four terms of consecutive bases: contribution to Skolem's conjecture

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