Question 142
Let $a \geq 2$ be a natural number. Show that the following statements are equivalent:
a) There exist nonzero natural numbers $b, c$ such that $a^{2}=b^{2}+c^{2}$;
b) There exists a nonzero natural number $d$ such that the equations $x^{2}-a x+d=0$ and $x^{2}-a x-d=0$ have integer roots.
a) There exist nonzero natural numbers $b, c$ such that $a^{2}=b^{2}+c^{2}$;
b) There exists a nonzero natural number $d$ such that the equations $x^{2}-a x+d=0$ and $x^{2}-a x-d=0$ have integer roots.