Are monochromatic Pythagorean triples unavoidable under morphic colorings

ABSTRACTA Pythagorean triple is a triple of positive integers a,b,c∈N+ satisfying a2 + b2 = c2. Is it true that for any finite coloring of N+, at least one Pythagorean triple must be monochromatic? In other words, is the Diophantine equation X2 + Y2 = Z2 regular? This problem, recently solved for 2-colorings by massive SAT computations [Heule et al., 2016], remains widely open for k-colorings with k ⩾ 3. In this article, 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.