Sum-of-Three-Cubes Problem Solved for ‘Stubborn’ Number 33 | Mathematics - Quanta Magazine

A number theorist with programming prowess has found a solution to 33 = x³ + y³ + z³, a much-studied equation that went unsolved for 64 years, according to John Pavlus, writer and filmmaker.

Photo: Lucy Reading-Ikkanda/Quanta Magazine

Mathematicians long wondered whether it’s possible to express the number 33 as the sum of three cubes — that is, whether the equation 33 = x³+ y³+ z³ has a solution. They knew that 29 could be written as 3³ + 1³ + 1³, for instance, whereas 32 is not expressible as the sum of three integers each raised to the third power. But the case of 33 went unsolved for 64 years.

Now, Andrew Booker, a mathematician at the University of Bristol, has finally cracked it: He discovered that (8,866,128,975,287,528)³ + (–8,778,405,442,862,239)³ + (–2,736,111,468,807,040)³ = 33.

Booker found this odd trio of 16-digit integers by devising a new search algorithm to sift them out of quadrillions of possibilities. The algorithm ran on a university supercomputer for three weeks straight. (He says he thought it would take six months, but a solution “popped out before I expected it.”) When the news of his solution hit the internet earlier this month, fellow number theorists and math enthusiasts were feverish with excitement...

With 33’s winning ticket now in hand, Booker plans to look next for a solution for 42. (He already determined that none exist in the ten-quadrillion range; he’ll have to look further out on the number line, to at least 10¹⁷.) But even when he or another number theorist has identified sum-of-three-cubes solutions for every eligible integer up to 100, they’ll then face 11 more “stubborn” integers without sum-of-three-cubes solutions between 101 and 1,000, and an infinitude of them beyond that. What’s more, Booker and other experts say, each new solution found for one of these holdouts sheds no theoretical light on where, or how, to find the next one. “I don’t think these are sufficiently interesting research goals in their own right to justify large amounts of money to arbitrarily hog a supercomputer,” Booker said.

So why bother with 33 or 42 at all?
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Source: Quanta Magazine