Joe's primes

Algebra Level 5

Let \(x, y\) be complex numbers satisfying

\[ \begin{align} x + y & = a, \\ xy &= b,\\ \end{align} \]

where \(a\) and \(b\) are positive integers from 1 to 100 inclusive. What is the sum of all possible distinct values of \(a\) such that \(x^3 + y^3\) is a positive prime number?

This problem is posed by Joe T.

Details and assumptions

It is stated that \(x\) and \(y\) are complex numbers. They need not be positive integers.


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