Let x,y be complex numbers satisfying
x+y=a
xy=b,
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?
To find the possible distinct values of a such that x^3+y^3 is a positive prime number, we need to analyze the expression x^3 + y^3 and its factorization.
First, let's use the identity for the sum of cubes: x^3 + y^3 = (x + y)(x^2 - xy + y^2).
Since we know that x + y = a and xy = b, we can substitute these values into the expression x^3 + y^3:
x^3 + y^3 = (x + y)(x^2 - xy + y^2) = a(x^2 - xy + y^2).
Now, we want to find the values of a such that x^3 + y^3 is a positive prime number. The expression x^3 + y^3 will be prime only if a is a prime number, and (x^2 - xy + y^2) is equal to 1.
To find the distinct values of a, we need to find the pairs of x and y that satisfy the equation x^2 - xy + y^2 = 1.
Let's rearrange the equation to get y^2 - xy + x^2 - 1 = 0.
This equation represents a quadratic equation in terms of y. Using the quadratic formula, we have:
y = (x ± sqrt(-3x^2 + 4))/2.
Notice that for y to be a complex number, the expression inside the square root, -3x^2 + 4, must be negative.
To find the range of x for which the expression inside the square root is negative, we solve the inequality -3x^2 + 4 < 0.
-3x^2 + 4 < 0
3x^2 - 4 > 0
(√3x + 2)(√3x - 2) > 0.
The solution to this inequality is x < -2/√3 or x > 2/√3.
Now, we need to find the values of x that satisfy this range and find their corresponding values of y. We can then calculate a = x + y for each pair (x, y).
By iterating through all possible values of x within the given range, we can find all the distinct values of a that satisfy the conditions stated in the problem. For each value of a, we check if x^3 + y^3 is a positive prime number.
By summing all the distinct values of a that satisfy the conditions, we can find the sum of all possible distinct values of a.