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Mathematician Cracks the 33 Problem

Mathematicians have been trying for 64 years to express the number 33 as the sum of three cubes. Andrew Booker, Reader of Pure Mathematics at the University of Bristol in the U.K., has cracked the equation, leaving the number 42 as the last number unsolved for three cubes. Wikimedia Commons

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If you're a trivia junkie, you may know of 33 as Kareem Abdul-Jabbar's old jersey number, or as the mysterious notation on bottles of Rolling Rock beer. If you make a lot of international phone calls, you might know that it's the country code for France.

Chances are, though, that unless you're really, really into 33, you probably don't know that mathematicians have been trying to figure out for the past 64 years whether it's possible to come up with 33 as the sum of three cubes (as an equation, it's 33 = x³+ y³+ z³). (For a more sophisticated explanation, try this Quanta Magazine article.)

It's an example of something called a Diophantine equation, in which all the unknowns must be integers, or whole numbers. With some numbers, this sort of thing is pretty easy. As Massachusetts Institute of Technology professor Bjorn Poonen explained in this 2008 paper, the number 29, for example, is the sum of the cubes of 3, 1 and 1. For 30, in contrast, the three cubes are all 10-digit numbers, and two of them are negative integers. Math is strange like that.

Expressing 33 as the sum of three cubes has proven devilishly elusive. That is, until recently. A solution was worked out by Andrew Booker, who holds a doctorate in mathematics from Princeton and is a reader (a research-oriented faculty position) in pure mathematics at the University of Bristol in the UK.

In this YouTube video from Numberphile, Booker explains that after he saw a video on the solving of the three cubes problem for 74, he got the inspiration to tackle 33:

Ultimately, he devised a new, more efficient algorithm than mathematicians had been using up to this point.

"It probably looks like I've made things a lot more complicated," he explained in the video, as he wrote out calculations on a big brown sheet of paper.

To crunch the numbers, he then used a cluster of powerful computers — 512 central processing unit (CPU) cores at the same time — known as Blue Crystal Phase 3. When he returned to his office one morning after dropping his children off at school, he spotted the solution on his screen. "I jumped for joy," he recalled.

The three cubes are 8,866,128,975,287,5283; - 8,778,405,442,862,2393; and -2,736,111,468,807,0403.

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