Robin Houston reports that 42 is the sum of three cubes. It is special in this respect, not because he is alone, but because it was for an obvious reason the last remaining number below 100 where the sum was not discovered.
42 = (-80538738812075974) ^ 3 + 80435758145817515 ^ 3 + 12602123297335631 ^ 3
This computational advance was achieved in a collaboration between Andrew Sutherland (MIT) and Andrew Booker (Bristol). They announced the result by replacing their home pages with the expression, with the title of the page Life, the universe and everything.
Each cube of an integer is within one of a multiple of nine, which means that a sum of three cubes must be within three of a multiple of nine. Therefore, the numbers of the form 9𝑘 + 4 or 9𝑘 + 5 cannot be written as the sum of three cubes.
In 1992, Roger Heath-Brown surmised that any other integer can be written as the sum of three cubes, in infinite different ways. Mathematicians in general seem to have been convinced by Heath-Brown's argument that this should be true, but actually finding ways to write any particular number as a sum of three cubes remains a difficult problem.
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