Boolean Pythagorean triples problem: Difference between revisions

Revision as of 09:25, 8 March 2021

Revision as of 09:26, 8 March 2021

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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 consist of all red or all blue members. 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>

==Statement==

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==Solution==

Marijn Heule, Oliver Kullmann and Victor W. Marek 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"/>
}}

There are {{nowrap|2⁷⁸²⁵ ≈ 3.63×10²³⁵⁵}} possible coloring combinations for the numbers up to [[7825 (number)|7825]]. These possible colorings were logically and algorithmically narrowed down to around a trillion (still highly complex) cases, and those were examined using a [[Boolean satisfiability]] solver. Creating the proof took about 4 CPU-years of computation over a period of two days on the Stampede supercomputer at the [[Texas Advanced Computing Center]] and generated a 200 terabyte propositional proof, which was compressed to 68 gigabytes.