FOM: Sum of three cubes

Joe Shipman shipman at
Fri Nov 5 12:42:33 EST 1999

Earlier, I asked for the smallest number known not to be a sum of three
cubes; I should have added the specification N mod 9 not in {4,5},
because all cubes are 0, 1, or -1 mod 9.The answer seems to be that NO
number is known not to be a sum of 3 cubes except for the trivial reason
of being 4 or 5 mod 9!  The first unknown case is 30.

If the conjecture (Conn and Vaserstein) "all numbers not 4 or 5 mod 9
are sums of 3 cubes" is true, then the set of numbers that are sums of 3
cubes is recursive.  Elkies reports numerical evidence for this;
heuristically one should expect infinitely many solutions.  See for details.

So Mazur's example, as repeated here by Matthew Frank, seems dubious as
a candidate for a natural nonrecursive non-r.e.complete set.  Other
similar sets (maybe the set of sums of three 5th powers, for example?)
do not seem to have this problem because heuristic arguments suggest
only finitely many solutions for permitted residue classes.

-- Joe Shipman

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