Question 1
Let $a \ge 2$ be a positive integer. Prove that the following statements are equivalent:
a) One can find positive integers $b, c$, such that $a^2 = b^2 + c^2$.
b) One can find a positive integer $d$, such that the equations $x^2 - a x + d = 0$ and $x^2 - a x - d = 0$ have integer roots.
a) One can find positive integers $b, c$, such that $a^2 = b^2 + c^2$.
b) One can find a positive integer $d$, such that the equations $x^2 - a x + d = 0$ and $x^2 - a x - d = 0$ have integer roots.