Boolean Pythagorean triples problem: Difference between revisions

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This problem is from [[Ramsey theory]] and asks if it is possible to color each of the positive integers either red or blue, so that no Pythagorean triple of integers ''a'', ''b'', ''c'', satisfying <math>a^2+b^2=c^2</math> are all the same color. For example, in the Pythagorean triple 3, 4 and 5 (<math>3^2+4^2=5^2</math>), if 3 and 4 are colored red, then 5 must be colored blue.

Marijn Heule, Oliver Kullmann and Victor Marek investigated the problem, and showed that such a coloring is impossible. Up to the number 7824 it isonly possible to color the numbers such that all Pythagorean triples are admissible, but the proof shows that no such coloring can be extendedup to also color the number 78257824. The actual statement of the theorem proved is

{{math theorem| The set {1, . . . , 7824} can be partitioned into two parts, such that no part contains a Pythagorean triple, while this is impossible for {1, . . . , 7825}.<ref name="arXiv"/>}}