For natural numbers $m,n$, set $a=(n+1)^m-n$ and $b=(n+1)^{m+3}-n$. (a) Prove that $a$ and $b$ are coprime if $m$ is not divisible by $3$. (b) Find all numbers $m,n$ for which $a$ and $b$ are not coprime.
Source: Croatia 1998 2nd Grade P4
Tags: number theory
For natural numbers $m,n$, set $a=(n+1)^m-n$ and $b=(n+1)^{m+3}-n$. (a) Prove that $a$ and $b$ are coprime if $m$ is not divisible by $3$. (b) Find all numbers $m,n$ for which $a$ and $b$ are not coprime.