Question 59
We say that a positive integer $n$ is fantastic, if there exist positive rational numbers $a$ and $b$ such that
$$
n = a + \frac{1}{a} + b + \frac{1}{b}.
$$
a. Prove that there exist infinitely many prime numbers $p$ such that no multiple of $p$ is fantastic.
b. Prove that there exist infinitely many prime numbers $p$ such that some multiple of $p$ is fantastic.
$$
n = a + \frac{1}{a} + b + \frac{1}{b}.
$$
a. Prove that there exist infinitely many prime numbers $p$ such that no multiple of $p$ is fantastic.
b. Prove that there exist infinitely many prime numbers $p$ such that some multiple of $p$ is fantastic.