Consider the following problem:
Given k, a natural number, determine if there exists x,y,z INTEGERS such that x3+y3+z3=k.
It is not obvious that this problem is decidable (I think it is but have not been able to find an exact statement to that affect; however, if it was not solvable, I would know that, hence it is solvable. If you know a ref give it in the comments.)
If k≡ 4,5 mod 9 then mod arguments easily show there is no solution. Huisman showed that if k≤ 1000, k≡1,2,3,6,7,8 mod 9 and max(|x|,|y|,|z|) ≤ 1015 and k is NOT one of
33, 42, 114, 165, 390, 579, 627, 633, 732, 795, 906, 921, 975
then there was a solution. For those on the list it was unknown.
Recently Booker (not Cory Booker, the candidate for prez, but Andrew Booker who I assume is a math-computer science person and is not running for prez) showed that
x3 + y3 + z3 =33
DOES have a solution in INTEGERS. It is
does that make us more likely or less likely to think that
x3 + y3 + z3 =42
has a solution? How about =114, etc, the others on the list?
Rather than say what I think is true (I have no idea) here is what I HOPE is true: that the resolution of these problems leads to some mathematics of interest.