The problem of 42 — at least as it relates to whether the number could be considered the sum of three cubes — has finally been solved. The question of whether every number under 100 could be expressed in this fashion has been a long-standing puzzle in the world of mathematics. Now, two mathematicians, Andrew Sutherland of MIT and Andrew Booker of Bristol, have jointly proven that 42 is indeed the sum of three cubes.
For years, mathematicians have worked to demonstrate that x3+y3+z3 = k, where k is defined as the numbers from 1-100. By 2016, researchers had demonstrated that this theory held true in all cases except for two unproven exceptions: 33 and 42. The formal theory, as expressed by Roger Heath-Brown in 1992, is that every k unequal to 4 or 5 modulo 9 has infinitely many representations as the sum of three cubes. By closing this particular gap, we’ve now proven that all numbers below 113 fit this theory.
Earlier this year, Andrew Booker of Bristol was inspired by a Numberphile video to begin working on a solution. We’ve embedded that video below:
Booker came up with a new, more efficient algorithm to search for a solution to the problem for these two values. The solution for 33 took about three weeks to find once the problem was run through a supercomputer at the UK’s Advanced Computing Research Centre. 42 proved a tougher nut to crack, so Booker paired up with Andrew Sutherland, who is an expert in massively parallel computation in addition to being a mathematician. The two enlisted the help of the Charity Engine, a distributed computing project that allows PCs to make money for charities through the donation of computing time.
Over a million hours of computation later, the team had its solution. In the equation x3+y3+z3 = k, let x = -80538738812075974, y = 80435758145817515, and z = 12602123297335631. Plug it all in, and you get (-80538738812075974)3 + 804357581458175153 + 126021232973356313 = 42. And with that, we’ve found solutions for all the values of k up to 100 (technically, up to 113).
“I feel relieved,” Booker said. “In this game, it’s impossible to be sure that you’ll find something. It’s a bit like trying to predict earthquakes, in that we have only rough probabilities to go by. So, we might find what we’re looking for with a few months of searching, or it might be that the solution isn’t found for another century.”
It may not prove that 42 is the Answer to the Ultimate Question of Life, the Universe, and Everything, but Douglas Adams clearly made the case for that solution in the mathematical and philosophical textbook, The Hitchhiker’s Guide to the Galaxy. Efforts to understand the Ultimate Question remain mired in disgruntled physics equations regarding the intrinsic difficulty of building planet-sized supercomputers with molten iron for a central core.
Top image credit: Martinultima/Wikipedia