Standard morphisms and Pythagorean triples

Abstract

Let $m\geq 1$, let $f:\mathbb N\to\mathbb Z/m\mathbb Z$ be a standard morphism and let $T(m)$ be the least integer $N$ such that every such $f$ admits a primitive monochromatic Pythagorean triple with hypotenuse at most $N$. The aim of this note is to prove that every standard morphism has infinitely many identity-valued Pythagorean triples and infinitely many primitive monochromatic Pythagorean triples. Thus the qualitative part of Problem~4.3 of Eliahou, Fromentin, Marion-Poty and Robilliard is solved for every $m$. Moreover, $T(m)$ is finite, the morphism $n\mapsto v_3(n)\pmod m$ has least possible hypotenuse $(9^m+1)/2$ and $T(d)\leq T(m)$ when $d\mid m$.