Let $n\geq 2$ be an integer. The permutation $a_1,a_2,..., a_n$ of the numbers $1, 2,...,n$ is called quadratic if $a_ia_{i +1} + 1$ is a perfect square for all $1\leq i \leq n-1.$ The permutation $a_1,a_2,..., a_n$ of the numbers $1, 2,...,n$ is called cubic if $a_ia_{i + 1} + 1$ is a perfect cube for all $1\leq i \leq n - 1.$ a) Prove that for infinitely many values of $n$ is there at least one quadratic permutation of the numbers $1, 2,...,n.$ b) Prove that for no value of $n$ is there a cubic permutation of the numbers $1, 2,..., n.$