AI 中文总结
本文证明每个标准态射有无穷多恒值毕达哥拉斯三元组及无穷多本原单色毕达哥拉斯三元组，解决相关问题的定性部分，还给出$T(m)$的有限性、特定态射的最小斜边及$T(d)$与$T(m)$的大小关系。
AI 中文摘要
设$m\geq1$，$f:\mathbb{N}\to\mathbb{Z}/m\mathbb{Z}$为标准态射，$T(m)$为满足：每个此类$f$都存在斜边不超过$N$的本原单色毕达哥拉斯三元组的最小整数$N$。本文旨在证明，每个标准态射都有无穷多个取恒等值的毕达哥拉斯三元组和无穷多个本原单色毕达哥拉斯三元组，由此解决了Eliahou、Fromentin、Marion-Poty及Robilliard所提问题4.3对任意$m$的定性部分；此外，$T(m)$是有限的，态射$n\mapsto v_3(n)\pmod{m}$的最小可能斜边为$(9^m+1)/2$，且当$d\mid m$时$T(d)\leq T(m)$。
英文摘要
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 $d\mid m$.
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