Lowest Common Multiple of Integers/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:

$\lcm \set {2 n - 1, 2 n + 1} = 4 n^2 - 1$

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

$\text {(1)}: \quad$

$\ds 2 n + 1$

$=$

$\ds 1 \times \paren {2 n - 1} + 2$

$\text {(2)}: \quad$

$\ds 2 n - 1$

$=$

$\ds 2 \times \paren {n - 1} + 1$

$\text {(3)}: \quad$

$\ds n - 1$

$=$

$\ds 1 \times \paren {n - 1}$

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

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

Thus:

$\ds \lcm \set {2 n - 1, 2 n + 1}$

$=$

$\ds \paren {2 n + 1} \paren {2 n - 1}$

$\ds $

$=$

$\ds \paren {2 n}^2 - 1$

The result follows.

$\blacksquare$

1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-2}$ Divisibility: Exercise $5 \ \text {(f)}$