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42 is the sum of three cubes / Boing Boing



42 is the sum of three cubes / Boing Boing

Robin Houston announces that 42 is the sum of three cubes. This is special in this respect not because he is alone, but because for the obvious reason the last remaining number was below 100, where the amount was not found.

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

This computational breakthrough was achieved in collaboration between Andrew Sutherland (MIT) and Andrew Booker (Bristol). They announced the result, replacing their homepages with the expression – with the title of the Life, Universe, and Everything page.

Each cube of an integer is within one of the multiples of nine, which means that the sum of three cubes must be within three of the multiples of nine. So numbers 9𝑘 + 4 or 9𝑘 + 5 cannot be written as a sum of three cubes.

In 1992, Roger Heath-Brown suggested that any other integer can be written as a sum of three cubes, in infinitely many different ways. In general, mathematicians seem to have been convinced of Heath-Brown's argument that this should be true, but in fact finding ways to write down any given number as a sum of three cubes remains a difficult problem.

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Rob Beschizza

Follow Rob @beschizza on Twitter.

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