Got a terabyte external hard drive at home? Great. Now go buy 199 more and you'll be able to store all the data that was required to make this math proof. Nature reports on a new paper in arXiv outlining how an international team of researchers created this enormous data-heavy proof.
The problem they were trying to prove one way or the other is a question dating back to the 1980s involving Pythagorean triples. If you don't remember the Pythogrean theorem from high school geometry, it's the one that says that for any right triangle, a2 + b2 = c2 (where a and b are the shorter sides and c is the longest one, the hypotenuse). There are certain sets of numbers where you can satisfy the theorem with whole integers. For example, 52 + 122 = 132.
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Got it? Now imagine that every integer is painted either red or blue. It's like there's an election coming and you have to pick one side or the other, and stay that way. The boolean Pythagorean triples problem, as put forth by mathematician Ronald Graham in the 1980s, asks whether, in this two-color scenario, you could color the numbers so that no set of Pythagorean triples are all the same color—that is, all three red or all three blue.
This is a trickier question than it might sound. What makes it so is that one integer can be part of multiple Pythagorean triples. Take 5. So 3, 4, and 5 are a Pythagorean triple. But so are 5, 12, and 13. If 5 is blue in the first example, then it has to be blue in the second, meaning either 12 or 13 has to be red. Carry this logic forward into much bigger numbers and you could see where this would start to get tricky. If 12 has to be red in that 5-12-13 triple, it might force changes down the line that would result in a monochrome triple somewhere.
In fact, it does. The answer to Graham's question——could one color the numbers so that no set of Pythagorean triples are all the same color—is no, but proving it requires a computer to work out via brute force calculations all of those combinations of numbers and colors. That's what the team did here, earning a big fat $100 check from Graham in the process. It turns out that there are solutions for the question up to 7,824, but as soon as you try to do it for 7,825, you fail. There are no solutions. Proving it took a dataset of 200 TB, blowing away the previous largest proof of 13 gigabytes.
Not everyone is so impressed, though. Says Nature:
Although the computer solution has cracked the Boolean Pythagorean triples problem, it hasn't provided an underlying reason why the coloring is impossible, or explored whether the number 7,825 is meaningful… That echoes a common philosophical objection to the value of computer-assisted proofs: they may be correct, but are they really mathematics?