Boolean Pythagorean triples problem

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

This problem is part of the Ramsey theory and asks if it is possible to color all the integers either red or blue so that no Pythagorean triple of integers a, b, c, satisfying $a^{2}+b^{2}=c^{2}$ are all the same color. For example if you would color a and b red, and c blue, this would successfully not satisfy the tested triple, but all triples would have to be tested.

The proof found out that up to the number 7,824 it is possible to color the numbers in such a way and that 10^2,300 colorings exist, but that for numbers above 7,825 no solutions exist. This means that for the problem question, we can say that no solution exists. The proof took two days of computer execution time on the Stampede supercomputer at the Texas Advanced Computing Center and generated 200 terabytes of data.

In the 1980s Ronald Graham offered a $100 prize for the solution of the problem, which has now been awarded to Marijn Heule. The paper describing the proof was published on arXiv on 3 May 2016.^[2] and has been accepted for the SAT 2016 conference.

References

^ Lamb, Evelyn (26 May 2016). "Two-hundred-terabyte maths proof is largest ever". Nature. doi:10.1038/nature.2016.19990.
^ Heule, Marijn J. H.; Kullmann, Oliver; Marek, Victor W. (2016-05-03). "Solving and Verifying the boolean Pythagorean Triples problem via Cube-and-Conquer". arXiv:1605.00723 [cs].

[1] Lamb, Evelyn (26 May 2016). "Two-hundred-terabyte maths proof is largest ever". Nature. doi:10.1038/nature.2016.19990.

[2] Heule, Marijn J. H.; Kullmann, Oliver; Marek, Victor W. (2016-05-03). "Solving and Verifying the boolean Pythagorean Triples problem via Cube-and-Conquer". arXiv:1605.00723 [cs].

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