Pieces of the Puzzle

So, were you and I waiting by the screen to see when we would finally know whether or not the number 33 could be expressed as the sum of three cubes?

Of course you weren't -- you, O Tolerant Reader, have a life and I have an inability to math very well. Nor, I imagine, did we know that there was a search on for that number. In any event, the solution to k = x³+ y³+ z³ when k = 33 has been found: (8,866,128,975,287,528)³ + (–8,778,405,442,862,239)³ + (–2,736,111,468,807,040)³ = 33.

For some numbers, the equation solves simply. You can write 29 as 3³ + 1³ + 1³. For others, there is no solution. Any number that has either 4 or 5 as a remainder when you divide it by 9 can't be written as the sum of three cubes. So while 33 has this newly-found solution, 32 will never have one. Divide 32 by 9 and you get 27, with 5 left over.

Andrew Booker of the University of Bristol wrote the algorithm which found the number. He figured the supercomputer running it would take six months to solve the problem, but it actually took only three weeks. There only two numbers between 1 and 100 that had never been solved were 33 and 42, and now only 42 remains. Booker will train his algorithm on that next, although the search will involve even larger numbers than the quadrillions that solved for 33.

One reason to find the answers to these so-called "stubborn numbers" is because mathematicians don't really like having unsolved equations laying around. Another is that finding solutions like this can play a role in some future attempts to find proofs for k = x³+ y³+ z³, or proofs that use it.

Left as yet undiscussed is the possibility that solving this polynomial for 42 might just be the way to find three cubes that add up to everything.