We all know that 42 is the answer to life, the universe, and everything, thanks to The Hitchhiker’s Guide to the Galaxy. Now, we also know that it’s the sum of three cubes.
For decades, scientists have wondered whether each of the numbers from 0 to 100 could be represented as the sum of three cubes, where a cube is the same number multiplied together three times (two cubed equals eight). Forty-two was the last number without a proven solution – until now.
“It’s awesome,” MIT mathematician Andrew Sutherland told Gizmodo. “You’re searching and hope it’s there and just don’t know if the algorithm is going to find it. You wait and wait and just when you’re at the point of giving up, the number shows up. It’s very gratifying.”
Researchers Andrew Sutherland at MIT and Andrew Booker at the University of Bristol in the United Kingdom found the result using over a million hours of computing time on the Charity Engine, according to a press release. The Charity Engine is a computing platform that takes unused processing power from 500,000 home computers to produce a kind of world-wide supercomputer.
(-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3 = 42
Mathematicians since Louis J. Mordell in the 1950s have been working toward solutions of the equation a3+b3+c3=n, where n is the number of interest (42 in this case) and a, b, and c are the solutions they’re hunting for. Scientists had found a, b, and, c for all numbers less than 100 except for proven exceptions that would have no solution, as well as 33 and 42.
Most of the exceptions come from a separate proof that all cubes are either multiples of nine or one integer away from a multiple of nine on the number line. That means that three cubes summed together could only result in numbers three or less units away from multiples of nine – you could never add three cubes up to a number four or five units away from a multiple of nine. But 33 and 42 were exceptions; both are three units away from multiples of nine, but neither had a proven solution. Mathematicians had figured that both numbers (and any numbers other than those explicitly forbidden) should have a solution, but there isn’t a proof explicitly saying so.
Motivated by a YouTube video on the topic, Booker produced an algorithm for finding a solution to these problems, and found a solution for n=33 earlier this year. Now he and Sutherland have found a solution to n=42 after months more effort.
“It’s like winning the lottery,” Sutherland said. “If you play long enough you’re guaranteed to win, but there’s no guarantee for how long it will take.”
There are now several numbers smaller than 1,000 without a sum of three cube solution, Sutherland explained, but he’s more interested in sums of three cubes that produce the number 3. Mathematicians have since proven that 1 and 2 have infinitely many solutions of a predictable pattern, but they’ve only found trivial, easy solutions for 3 (1 cubed + 1 cubed + 1 cubed = 3, for example). They’re still wondering when another, larger-number solution will turn up.
If this seems like mathematic frivolity, it’s not. These Diophantine equations, where you have to figure out several unknowns that combine into a known value, are used throughout computing in various algorithms. But what these researchers are really doing, finding points on elliptic curves, is a fundamental mathematical idea used in the cryptography that secures things like bitcoin.
But if you don’t care about any of that, just know that the answer to the life, the universe, and everything now has another properly absurd question to go along with it.
Featured photo: University of Bristol