Are monochromatic Pythagorean triples unavoidable under morphic colorings ?

Abstract : 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.
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Submitted on : Thursday, March 30, 2017 - 2:24:39 PM
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  • HAL Id : hal-01309134, version 2
  • ARXIV : 1605.00859

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S Eliahou, Jean Fromentin, V Marion-Poty, D Robilliard. Are monochromatic Pythagorean triples unavoidable under morphic colorings ?. 2016. ⟨hal-01309134v2⟩

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