Are monochromatic Pythagorean triples unavoidable under morphic colorings ?

TL;DR

This paper investigates whether monochromatic Pythagorean triples are unavoidable under certain structured colorings of positive integers, extending known results for 2-colorings to 3-colorings using morphic colorings.

Contribution

It introduces morphic colorings with multiplicative properties and demonstrates their role in guaranteeing monochromatic Pythagorean triples in small intervals for 2 and 3 colors.

Findings

Monochromatic Pythagorean triples are unavoidable in many morphic colorings.

Such triples appear in small integer intervals under these colorings.

The study extends understanding of colorings beyond 2-color cases.

Abstract

A Pythagorean triple is a triple of positive integers a, b, c $\in$ N${}^{+}$ satisfying a${}^2$ + b${}^2$ = c${}^2$. Is it true that, for any finite coloring of N${}^{+}$ , at least one Pythagorean triple must be monochromatic? In other words, is the Dio-phantine equation X${}^2$+ Y${}^2$ = Z${}^2$ regular? This problem, recently solved for 2-colorings by massive SAT computations [Heule et al., 2016], remains widely open for k-colorings with k $\ge$ 3. In this paper, we introduce morphic colorings of N + , which are special colorings in finite groups with partly multiplicative properties. We show that, for many morphic colorings in 2 and 3 colors, monochromatic Pythagorean triples are unavoidable in rather small integer intervals.