Boolean Pythagorean triples problem: Difference between revisions

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{{short description|Can one split the integers into two sets such that every Pythagorean triple spans both?}}

ThisThe '''Boolean Pythagorean triples problem''' is a problem from [[Ramsey theory]] andabout [[Pythagorean triple]]s, which 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.

The '''Boolean Pythagorean triples problem''' is a problem relating to [[Pythagorean triple]]s which was solved using a [[computer-assisted proof]] in May 2016.<ref name="nature">{{Cite journal|last=Lamb|first=Evelyn|date=26 May 2016|title=Two-hundred-terabyte maths proof is largest ever|url=http://www.nature.com/news/two-hundred-terabyte-maths-proof-is-largest-ever-1.19990|journal=Nature|doi=10.1038/nature.2016.19990|volume=534|pages=17–18|pmid=27251254|bibcode=2016Natur.534...17L|doi-access=free}}</ref>

The '''Booleanproblem Pythagoreanwas triplessolved problem'''by isMarijn aHeule, problem relatingOliver toKullmann and [[PythagoreanVictor tripleW. Marek]]s whichin wasMay solved2016 usingthrough a [[computer-assisted proof]] in May 2016.<ref name="nature">{{Cite journal|last=Lamb|first=Evelyn|date=26 May 2016|title=Two-hundred-terabyte maths proof is largest ever|url=http://www.nature.com/news/two-hundred-terabyte-maths-proof-is-largest-ever-1.19990|journal=Nature|doi=10.1038/nature.2016.19990|volume=534|pages=17–18|pmid=27251254|bibcode=2016Natur.534...17L|doi-access=free}}</ref> They showed that such a coloring is only possible up to the number 7824. The actual statement of the theorem proved is

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 W. Marek]] investigated the problem, and showed that such a coloring is only possible up to the number 7824. 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"/>}}