Question 52
Let $a, b$ be integers, and let $P(x) = a x^{3} + b x$. For any positive integer $n$ we say that the pair $(a, b)$ is $n$-good if $n \mid P(m) - P(k)$ implies $n \mid m - k$ for all integers $m, k$. We say that $(a, b)$ is very good if $(a, b)$ is $n$-good for infinitely many positive integers $n$.
a. Find a pair $(a, b)$ which is $51$-good, but not very good.
b. Show that all $2010$-good pairs are very good.
(Turkey)
a. Find a pair $(a, b)$ which is $51$-good, but not very good.
b. Show that all $2010$-good pairs are very good.
(Turkey)