In their paper uploaded to the preprint server arXiv, Marijn Heule with the University of Texas, Oliver Kullmann with Swansea University and Victor Marek with the University of Kentucky have named the math problem as Boolean Pythagorean Triples problem. It was first proposed back in the 1980’s by mathematician Ronald Graham. He also offered a prize of US$100 for anyone who could solve it. Earlier this month, he duly presented the cheque to one of the three computer scientists, Marijn Heule. The problem asks if it is possible to colour each positive integer either red or blue, so that no trio of integers a, b and c satisfy Pythagoras’ famous equation a2 + b2 = c2 and all are the same colour. For instance, for the Pythagorean triple 3, 4 and 5, if 3 and 5 were coloured blue, 4 would have to be red. The researchers applied the Cube-and-Conquer paradigm, which is a hybrid of the SAT method for difficult problems to resolve the problem. It uses both look-ahead methods and CDCL solvers. Before handing the problem over to the computer, the researchers also did some of the math on their own by using many methods to cut down the number of choices to just one trillion (from 102,300) for the supercomputer to check. It took the team about 2 days running 800 processors in parallel on the University of Texas’s Stampede supercomputer to zip through all the possibilities. The researchers then verified the proof using another computer program. Finally, the computer proof did give a definitive answer that there are many allowable ways to colour the integers up to 7,824, but after that you reach 7,825, it is impossible for every Pythagorean triple to be multi-coloured. The team technically along with their computer did create a proof for the Boolean Pythagorean triples problem, but the questions remain as to why the colouring is impossible, or explored whether the number 7,825 is meaningful, says Kullmann. That resonance a common philosophical objection to the value of computer-assisted proofs: they may be correct, but are they really mathematics? The study has been published on Nature.
