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On the length of an arithmetic progression of the form ${3^x+2^y}$

Hongnan Chen, Fenglin Huang, Sihui Zhang

Abstract

The conclusion that the length of an arithmetic progression of the form ${3^x+2^y}$ is at most six is proved.

On the length of an arithmetic progression of the form ${3^x+2^y}$

Abstract

The conclusion that the length of an arithmetic progression of the form is at most six is proved.
Paper Structure (6 sections, 13 theorems, 123 equations, 1 figure)

This paper contains 6 sections, 13 theorems, 123 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm 1.1']}
  3. The properties of $d$
  4. The type of ${a_1}, \dots, {a_4}$
  5. ${a_1}, \dots, {a_4}$ Specific discussion
  6. Appendices

Key Result

Proposition 1.1

(cf. [1]) There are exactly five positive integers that can be written in multiple ways as the sum of non-negative powers of 2 and non-negative powers of 3, and there are only five integers that can be expressed in two ways as ${3^a+2^b\left( a,b\in N\right) }$: 5, 11, 17, 35, 259. The five element

Figures (1)

  • Figure 1: program

Theorems & Definitions (14)

  • Proposition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.9
  • ...and 4 more