# Boolean Pythagorean triples problem

The **Boolean Pythagorean triples problem** is a problem relating to Pythagorean triples which was solved using a computer-assisted proof in May 2016.[1]

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 are all the same color. For example, in the Pythagorean triple 3, 4 and 5 (), 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 only possible up to the number 7824. The actual statement of the theorem proved is

**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}.[2]

There are 2^{7825} ≈ 3.63×10^{2355} colorings for the numbers up to 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.

The paper describing the proof was published on arXiv on 3 May 2016,[2] and has been accepted for the SAT 2016 conference, where it won the best paper award.[3]

In the 1980s Ronald Graham offered a $100 prize for the solution of the problem, which has now been awarded to Marijn Heule.[1]

## See also

## References

- Lamb, Evelyn (26 May 2016). "Two-hundred-terabyte maths proof is largest ever".
*Nature*.**534**: 17–18. Bibcode:2016Natur.534...17L. doi:10.1038/nature.2016.19990. PMID 27251254. - Heule, Marijn J. H.; Kullmann, Oliver; Marek, Victor W. (2016-05-03). "Solving and Verifying the Boolean Pythagorean Triples problem via Cube-and-Conquer".
*Lecture Notes in Computer Science*: 228–245. arXiv:1605.00723. doi:10.1007/978-3-319-40970-2_15. - SAT 2016