Let $a$ and $b$ be positive integers such that $a^2 + b^2$ is a prime number. Prove that...

Question

When you observe carefully, the question does not have much factorization that you can do. Do you have any rearrangements in mind?

Let a and b be positive integers such that $a^2 + b^2$ is a prime number. Prove that the equation $x^2 + ax + b + 1 = 0$ does not have integer roots.

Playing with the discriminant also does not seem to work.

Answer 1

Hint: Try to prove $a^2 + b^2$ can be written as the product of two factors.