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After 65 Years, Supercomputers Finally Solve This Unsolvable Math Problem

Popular Mechanics logo Popular Mechanics 6 days ago David Grossman
a flock of birds flying in the sky: For decades, this math problem has stumped the smartest mathematicians in the world. It took a supercomputer and millions of hours to finally solve it. © Andrew Daniels For decades, this math problem has stumped the smartest mathematicians in the world. It took a supercomputer and millions of hours to finally solve it.
  • Two more answers for a complex math problem have been found.
  • Called the "summing of three cubes," the challenge is to find x, y, and z.
  • It took over a million computing hours to find the solution.

For decades, a math puzzle has stumped the smartest mathematicians in the world. x3+y3+z3=k, with k being all the numbers from one to 100, is a Diophantine equation that's sometimes known as "summing of three cubes."

When there are two or more unknowns, as is the case here, only the integers are studied. The trick is finding integers that work for all equations, or the numbers for x, y, and z that will all equal k. Over the years, scientists have solved for nearly every integer between 0 and 100. The last two that remained were 33 and 42.

Here's a Numberphile video explaining why this problem has proved to be so tricky:

Replay Video

Earlier this year, Andrew Booker of the University of Bristol spent weeks with a supercomputer to finally arrive at a solution for 33. But 42, which by coincidence is a well-known number in pop culture, proved to be much more difficult.

So Booker turned to MIT math professor Andrew Sutherland, and Sutherland in turn enlisted the help of Charity Engine, which utilizes idle, unused computing power from over 500,000 home PCs to create a crowdsourced and environmentally conscious supercomputer.

The answers took over a million hours to compute. Without further ado, they are:

X = -80538738812075974, Y = 80435758145817515, and Z = 12602123297335631.

Well, obviously.

"I feel relieved," Booker says of breaking the 65-year old puzzle first set down at Cambridge in a press statement. "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."

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