# Computer cracks 200-terabyte maths proof

## The Boolean Pythagorean triples problem took 800 processors at the University of Texas Stampede supercomputer two days to solve. Viviane Richter reports.

Ever struggled writing proofs in algebra class? Here’s one you wouldn’t even hope to read.

While solving a decade-old puzzle, a team of US and UK scientists has created the world’s largest maths proof – a file that takes up a massive 200 terabytes.

That’s about the size of the entire digitised text held by the US Library of Congress and smashes the previous record which racked up a humble 13 gigabyte.

The so-called Boolean Pythagorean triples problem has puzzled mathematicians since the 1980s. It’s about Pythagoras’ famous theorem which relates the length of the three sides of a right triangle:

a2 + b2 = c2

There are certain sets of numbers which satisfy the theorem with whole integers for a, b and c, for instance:

32 + 42 = 52

Now imagine all integers had to pick a colour – red or blue.

The Boolean Pythagorean triples problem asks: is it possible to colour each positive integer either red or blue so that no trio of integers a, b and c that satisfy Pythagoras’ equation are all the same colour?

In the above triple 3, 4 and 5, for example, at least one integer would have to be red and at least one blue. And this rule would have to hold for every other Pythagorean triple in your colouring scheme.

The team found all triples could be multi-coloured in integers up to 7,824. As soon as they hit 7,825, it became impossible.

But to prove a solution doesn’t exist, you need to try all possibilities. There are more than 102,300 ways to colour all those integers, so the scientists used a few mathematical tricks to reduce the number of combinations to trial to just under one trillion.

Two days later, with 800 processors at the University of Texas Stampede supercomputer crunching all possibilities in parallel, the team had their answer – no.

There is no way to colour the integers 1 to 7,825 in a way that leaves all Pythagorean triples multi-coloured, the team reported in arXiv.

University of California, San Diego mathematician Ronald Graham told Nature “200 terabytes are unbelievable”. In the 1980s, he offered a \$100 prize to who could solve the Boolean Pythagorean triples problem.

While the authors have now cashed that prize, they were quick to point out that even though they solved the famous problem, they still haven’t solved the reason for why 7,825 is special.

Still not convinced? The authors have compressed their solution into a 68-gigabyte version you can download, reconstruct and verify.

If you have a spare 30,000 hours of processor time, that is.

Viviane Richter is a freelance science writer based in Melbourne.
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