Lowest Common Multiple/Examples/2n-1 and 2n+1

Example of Lowest Common Multiple of Integers
Let $n \in \Z_{>0}$ be a strictly positive integer. The lowest common multiple of $2 n - 1$ and $2 n + 1$ is:
 * $\lcm \set {2 n - 1, 2 n + 1} = 4 n^2 - 1$

Proof
We find the greatest common divisor of $2 n - 1$ and $2 n + 1$ using the Euclidean Algorithm:

Thus $\gcd \set {2 n - 1, 2 n + 1} = 1$.

Hence by definition $n$ and $n + 1$ are coprime.

Thus:

The result follows.