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arXiv · 1605.00859

Are monochromatic Pythagorean triples unavoidable under morphic colorings ?

S Eliahou·J Fromentin·V Marion-Poty·D Robilliard
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Abstract

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.

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S Eliahou, J Fromentin, V Marion-Poty, D Robilliard. 2017-04-24. Are monochromatic Pythagorean triples unavoidable under morphic colorings ?. https://doi.org/10.1080/10586458.2017.1306465

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