A Super Computer for a Super-long Proof

Some maths is just too hard or complicated for humans. Some would just take too long. A new proof for the Boolean Pythagorean triples problem is too long even to be read by a human. At 200TB (or 46,600 DVDs worth!) it is the longest mathematical proof ever – even the shortened version (still a hefty 68 GB) would take 30,000 hours for a computer to process.

The Boolean Pythagorean triples problem asks whether it is possible to assign all of the natural numbers (positive integers) a colour of either blue or red, so that the numbers in no Pythagorean triple are all the same colour. A Pythagorean triple is a set of three integers that satisfies the condition a2 + b2 = c2. This equation is useful for finding the lengths of the sides of a right angled triangle.

The problem is harder than it looks. Some integers are part of many Pythagorean triples, so for example 5 is in the triple 3, 4, 5 and 5, 12, 13. If 5 is red in the first triple, it stays red in the second, and either 12 or 13 or both must be blue. This continues into the much higher numbers, and the scientists, Marijn J. H. Heule, Oliver Kullmann and Victor W. Marek, have shown that this eventually breaks down 7,825.

They used the Cube-and-Conquer program, which searches all the possible solutions by placing them into cubes. They used the University of Texas’ Stampede Supercomputer, which ran 800 processors for two days to find the solution. There are 102300 possible ways to colour the integers up to 7,825, and even after a load of mathematical tricks from symmetry to number theory techniques were used there were a trillion combinations to check.

Even now the problem has been solved it doesn’t add much to our understanding of the problem, begging the question ‘is this a real proof?’ Previous computer proofs have later been verified by hand, improving our understanding. In this case, we still don’t understand why colouring the integers works until 7,824 but then stops at 7,825, so we can’t solve similar problems any more easily, meaning we’ve gained little more than a useless fact.

Sources:

Marijn J. H. Heule, Oliver Kullmann, Victor W. Marek, Cornell University Library, 3 May 2016, Solving and Verifying the boolean Pythagorean Triples problem via Cube-and-Conquer, http://arxiv.org/abs/1605.00723 (accessed 01/06/2016)

Evelyn Lamb, Nature News, 26th May 2016, Two-hundred-terabyte maths proof is largest ever, http://www.nature.com/news/two-hundred-terabyte-maths-proof-is-largest-ever-1.19990#/b2 (accessed 01/06/2016), Nature, 534, 17–18, 2nd June 2016

Jamie Condliffe, Gizmodo,  27 May 2016, The World’s Biggest Ever Maths Proof is a Whopping 200TB, http://www.gizmodo.co.uk/2016/05/the-worlds-biggest-ever-maths-proof-is-a-whopping-200tb/ (accessed 01/06/2016)