42
Why is 42 the answer to "the ultimate question of life, the universe, and everything"? Is it really an elusive number? Why was it so computationally hard to crack the sum-of-three-cubes puzzle for number 42?
Two mathematicians, Andrew Sutherland of MIT and Andrew Booker of Briston University, with their programing prowess solved the sum-of-three-cubes puzzle for 42 on September 6, 2019.
Booker had previosly cracked 33, the other remaining unsolved number between 0 - 100. Sutherland and Booker collaborated on devising an algorithm and with the computing assitance from UK-based Charity Engine, they were able to tap into the computing power of 400K volunteer's home PCs to solve this puzzle.
42 #
Diophantine equations are polynomial equations whose unknown variables must take integer values. The sum-of-three-cubes problem, stated as k = x^3 + y^3 + z^3, is what number theorists call a Diophantine equation.
“There is a single integer parameter, d, that determines a relatively small set of possibilities for x, y, and z such that the absolute value of z is below a chosen search bound B,” says Sutherland. “One then enumerates values for d and checks each of the possible x, y, z associated to d. In the attempt to crack 33, the search bound B was 1016, but this B turned out to be too small to crack 42; we instead used B = 1017 (1017 is 100 million billion).
The Charity Engine application, which is based on the Berkeley Open Infrastructure for Network Computing (BIONIC), uses a tiny fraction of the CPU resource available which would go otherwise unused. As a result, the carbon footprint of this computation — related to the electricity our computations caused the PCs in the network to use above and beyond what they would have used, in any case — is lower than it would have been if we had used a supercomputer.” Couple that technology with the mathematicians who are also really good programmers. Voila!
Sum-of-three-cubes: k = x^3 + y^3 + z^3
42 = (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3
While the mathematicians are out to crack the Diophantine equation for the next lowest unsolved integer, 114, over the next year, I intend to find out why 42 is the ultimate answer! :)
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