Let $n$ be a positive integer. a) Prove that the number $(2n + 1)^3 - (2n - 1)^3$ is the sum of three perfect squares. b) Prove that the number $(2n+1)^3-2$ is the sum of $3n-1$ perfect squares greater than $1$.
Source: 2001 Cuba MO 1.3
Tags: number theory, Perfect Squares
Let $n$ be a positive integer. a) Prove that the number $(2n + 1)^3 - (2n - 1)^3$ is the sum of three perfect squares. b) Prove that the number $(2n+1)^3-2$ is the sum of $3n-1$ perfect squares greater than $1$.