Joe's primes

Algebra Level 5

Let x,yx, y be complex numbers satisfying

x+y=a,xy=b, \begin{aligned} x + y & = a, \\ xy &= b,\\ \end{aligned}

where aa and bb are positive integers from 1 to 100 inclusive. What is the sum of all possible distinct values of aa such that x3+y3x^3 + y^3 is a positive prime number?

This problem is posed by Joe T.

Details and assumptions

It is stated that xx and yy are complex numbers. They need not be positive integers.

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