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?}}

The '''Boolean Pythagorean triples problem''' is a problem from [[Ramsey theory]] about whether the [[natural number|positive integers]] can be colored red and blue so that no [[Pythagorean triple]]s are the same color. The Boolean Pythagorean triples problem was solved by Marijn Heule, Oliver Kullmann and [[Victor W. Marek]] in May 2016 through a [[computer-assisted proof]].<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 '''Boolean Pythagorean triples problem''' is a problem from [[Ramsey theory]] about [[Pythagorean triple]]s. The problem 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.

==Statement==

The Boolean Pythagorean triples problem was solved by Marijn Heule, Oliver Kullmann and [[Victor W. Marek]] in May 2016 through a [[computer-assisted proof]].<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 '''Boolean Pythagorean triples problem''' is a problem from [[Ramsey theory]] about [[Pythagorean triple]]s. The problem 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.

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.

==Solution==