$a, b$ are positive whole numbers such that $a + b = a/b + b/a$. How many possible values can $a^4 + b^4$ be? $a, b$ are positive whole numbers such that $a + b = a/b + b/a$. How many possible values can $a^4 + b^4$ be?
I tried using $(a^2 + b^2)^2$ but I don't know what to do after.

You have $a^2-ba^2+ba-b^2=0$ then $a=\dfrac{-b\pm\sqrt{b^2-4(1-b)(-b^2)}}{2(1-b)}$ so that $5-4b$ must be a square. The only possible value is $b=1$ thus $a=1$ and $a^4+b^4=2$
