  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.
