On Double 3-Term Arithmetic Progressions

TL;DR

This paper investigates whether increasing positive integer sequences with bounded gaps necessarily contain a specific type of double 3-term arithmetic progression, exploring variations and related properties using computational methods.

Contribution

It introduces new results on the existence of double 3-term arithmetic progressions in bounded-gap sequences and discusses related properties using a specialized scripting language.

Findings

Sequences with bounded gaps may or may not contain double 3-term APs.

Several variations of the problem are analyzed.

Computational experiments support the theoretical findings.

Abstract

In this note we are interested in the problem of whether or not every increasing sequence of positive integers $x_1x_2x_3...$ with bounded gaps must contain a double 3-term arithmetic progression, i.e., three terms $x_i$, $x_j$, and $x_k$ such that $i + k = 2j$ and $x_i + x_k = 2x_j$. We consider a few variations of the problem, discuss some related properties of double arithmetic progressions, and present several results obtained by using RamseyScript, a high-level scripting language.