Are monochromatic Pythagorean triples avoidable?

S Eliahou (1) , J Fromentin (1) , V Marion-Poty (2) , Denis Robilliard (2)

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

A Pythagorean triple is a triple of positive integers a, b, c ∈ N-{0} satisfying a^2 + b^2 = c^2. Is it true that, for any finite coloring of N-{0}, at least one Pythagorean triple must be monochromatic? In other words, is the Diophantine equation X^2 +Y^2 = Z^2 regular? This problem has been open since several decades, even restricted to 2-colorings. In this note, we introduce partial morphisms, which are special colorings of N-{0} by finite groups with partly multiplicative properties. We show that, for many such colorings, monochromatic Pythagorean triples are impossible to avoid in the integer interval [1, 10000].

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

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

S Eliahou, J Fromentin, V Marion-Poty, Denis Robilliard. Are monochromatic Pythagorean triples avoidable?. 2016. ⟨hal-01309134v1⟩