Question 92
For each rational number $r$ consider the statement: If $x$ is a real number such that $x^2 - rx$ and $x^3 - rx$ are rational numbers, then $x$ is rational as well.
a) Prove the statement for $r \ge \frac{4}{3}$ and for $r \le 0$.
b) Let $p, q$ be different odd primes such that $3p \lt 4q$. Show the statement is false for $r = \frac{p}{q}$.
a) Prove the statement for $r \ge \frac{4}{3}$ and for $r \le 0$.
b) Let $p, q$ be different odd primes such that $3p \lt 4q$. Show the statement is false for $r = \frac{p}{q}$.