Are monochromatic Pythagorean triples unavoidable under morphic colorings ?

Shalom Eliahou (1) , Jean Fromentin (1) , Virginie Marion-Poty (2) , Denis Robilliard (2, 3)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville
2 LISIC - Laboratoire d'Informatique Signal et Image de la Côte d'Opale
3 ULCO - Université du Littoral Côte d'Opale

A 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 Dio-phantine 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 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.

Mots clés: Pythagorean triple; partition-regular equation; partial morphism; SAT solver

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]; Théorie des nombres [math.NT]

Shalom Eliahou, Jean Fromentin, Virginie Marion-Poty, Denis Robilliard. Are monochromatic Pythagorean triples unavoidable under morphic colorings ?. Experimental Mathematics, 2018, 27 (4), pp.419-425. ⟨10.1080/10586458.2017.1306465⟩. ⟨hal-01309134v2⟩