# Joe's primes

Algebra Level 5

Let $x, y$ be complex numbers satisfying

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

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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