FOM: Sum of three cubes

[Matthew Frank]

Thanks to Joe Shipman for the web reference on the sum of three cubes.
And hence a retreat from that conjecture:

I used to think that there were two interesting and contradictory
conjectures:  "all numbers not +/- 4 mod 9 are sums of three cubes" and
"the set of sums of three cubes is not computable"; even if the first was
more plausible than the second, my literature searches and conversations
with number theorists hadn't turned up any particularly good arguments
against the second, and so it still seemed worth investigating.  Now
Elkies's comments convince me that the latter conjecture is so implausible
as not to be worth investigating after all.  I would still like to find
some other "number-theoretically defined" set (in the sense of my previous
post) which is not computable, but I no longer know any particularly good
candidates.

--Matt Frank