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'Almost impossible' 66-year-old maths puzzle solved

Scatter plot of the running times of the computation. Each dot corresponds to one job executed by a computer in the Charity Engine's network.

An Electronic Delay Storage Automatic Calculator (EDSAC) - an early British computer which was used in the inaugural computation of the three cubes problem. Computer Laboratory, University of Cambridge

19 September 2019

The sum-of-three-cubes for the number 3 has been solved by Professor Andrew Booker and MIT's Professor Andrew Sutherland.

First they solved the Sum of Three Cubes for the difficult number 33, then for the really difficult 42, and now the team led by Andrew Booker of Bristol University and Andrew Sutherland of MIT has discovered a mind-bogglingly vast solution for the insanely difficult number that had kept mathematicians guessing since 1953; the number 3.

Despite having two trivial solutions (1,1,1 and 4,4,-5) nobody really knew if there were any more – and if there were, they would be almost unimaginably difficult to find. Years of searching had turned up nothing, until now.

The numbers required are an astonishing 21 digits long and 7000x bigger than those required for the solution to 42 discovered earlier this month. Despite this huge increase, Booker and Sutherland used ingenious new algorithms to reduce the search space to something more manageable, and which could be successfully explored, once again, by the Charity Engine planetary computer grid of 500,000+ home PCs.

The first non-trivial solution for 3 therefore required ‘only’ four million compute-hours to reveal itself:

569936821221962380720^3 + (-569936821113563493509)^3 + (-472715493453327032)^3 = 3

Booker and Sutherland hope to crack even more ‘almost impossible’ math problems using Charity Engine over the coming weeks.

Further information

‘Cracking the problem with 33’ by A. Booker in Research in Number Theory

Further information:
‘Sum-of-Three-Cubes Problem Solved for ‘Stubborn’ Number 33’ in Quanta Magazine [March 26 2019]

‘220,000 cores and counting: MIT math professor breaks record for largest ever Compute Engine job’ published on Google Cloud Platform Blog [April 20 2017]

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