Question 104
Consider the expression $M(n, m) = |n\sqrt{n^2 + a} - bm|$, where $n$ and $m$ are arbitrary positive integers and the numbers $a$ and $b$ are fixed; moreover, $a$ is an odd positive integer, and $b$ is a rational number with an odd denominator in its representation as an irreducible fraction.
Prove that there is
a) no more than a finite number of pairs $(n, m)$ for which $M(n, m) = 0$;
b) a positive constant $C$ such that the inequality $M(n, m) \ge C$ holds for all pairs $(n, m)$ with $M(n, m) \ne 0$.
Prove that there is
a) no more than a finite number of pairs $(n, m)$ for which $M(n, m) = 0$;
b) a positive constant $C$ such that the inequality $M(n, m) \ge C$ holds for all pairs $(n, m)$ with $M(n, m) \ne 0$.