Boolean Pythagorean triples problem

The Boolean Pythagorean triples problem is a problem from Ramsey theory about whether the positive integers can be colored red and blue so that no Pythagorean triples 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.

[1] 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

Marijn Heule, Oliver Kullmann and Victor W. Marek showed that such a coloring is only possible up to the number 7824.

These possible colorings were logically and algorithmically narrowed down to around a trillion (still highly complex) cases, and those, expressed as Boolean satisfiability problems, were examined using a SAT 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.

[3] The figure below shows a possible family of colorings for the set {1,...,7824} with no monochromatic Pythagorean triple, and the white squares can be colored either red or blue while still satisfying this condition.

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