OK, I couldn't resist the headline. The truth is that 42 is the sum of three cubes. If it doesn't sound as exciting to you, then all I can say is that you don't understand - yet.

Back in March of this year (2019) I reported on the finding of three cubes that sum to 33. The three cubes problem is a generalization of the Fermat problem:

Find three cubes, possibly negative, that sum to a given integer. That is, does:

x³+y³+z³=n

with x,y and z integers, possibly negative, add to n, a positive integer, not necessarily a cube.

It is very easy to state and easy to demonstrate. For example:

1³+2³+3³=36

So 36 is the sum of three cubes. The problem is that this isn't just about knowing that one particular number is the sum of three cubes, the conjecture is that all integers can be expressed as the sum of three cubes. The main part of the answer is very easy to solve. You cannot find three cubes that add to any number that is 4 or 5 mod 9 - i.e. if you divide by 9 and get remainder 4 or 5 then the number cannot be expressed as the sum of three cubes.

So we know that not every number can be expressed as the sum of three cubes, but what about the ones that are not 4 or 5 mod 9?

Solutions for this have been the subject of computer search since 1955 and for values less than 100 the question was open for only two numbers, 33 and 42. In March Andrew Booker of the University of Bristol settled the n=33 case:

(8,866,128,975,287,528)³ + (–8,778,405,442,862,239)³ +
                                                (–2,736,111,468,807,040)³ = 33

Now he, in conjunction with MIT's Drew Sutherland, has discovered the result:

( -80538738812075974)³ + (80435758145817515)³+
                                                     (12602123297335631)³ =42

Thus, we can now state that all of the integers up to 100 are expressible as the sum of three cubes.

That the final case is 42 brings a smile to the face of fans of the Douglas Adams Hitchhiker's Guide to the Galaxy (HGTTG), and if you haven't read it you must. If you don't know why, the reason is that in the book a planet-sized computer called Deep Thought was constructed to answer the ultimate question and after millennia of computing it answered "42". So the fact that 42 was the last holdout on the three cubes question was a very good joke indeed - unless it wasn't a joke and 42 is indeed the answer to Life the Universe and everything.

What might bring an even bigger smile to a HGTTG fan is that, yes a planetary-sized computer was used to solve the problem. OK, again I stretch the point with "planetary-sized computer". What was actually used was the Charity Engine a worldwide distributed computer which made use of idle time on over 500,000 home computers. The answer took over a million hours and, at the advertised price for 1 hour of CPU time, should have cost around $10,000.

The Charity Engine is a strange idea, the more so if you are familiar with similar schemes. It is based on the reasonably well known BOINC software, which drives the better known SETI@home project, which uses idle cpu time to search for signals from extraterrestrials. In this case the idea is that corporations can use the Charity Engine for 1 cent per hour of computing time and half the cash is donated to charity. The other half goes in randomly distributed prizes to the people providing the computers. If there are no corporate customers then the Charity Engine is used to provide computing power to volunteer computing projects. 

Numberpile interviewed Andrew Booker after the discovery:

If you want to join in the remaining unsolved cases up to a thousand are 114, 165, 390, 579, 627, 633, 732, 906, 921, and 975.

More Information

Cracking the problem with 33

https://www.charityengine.com

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