Thabit's Rule

Theorem
Let $n$ be a positive integer such that:

are all prime.

Then:
 * $\left({2^n a b, 2^n c}\right)$

forms an amicable pair.

Proof
Let $r = 2^n a b, s = 2^n c$.

Let $\sigma \left({k}\right)$ denote the sigma function on an integer $k$.

From Sigma of Power of 2:
 * $\sigma \left({2^n}\right) = 2^{n + 1} - 1$

From Sigma of Prime Number:

From Sigma Function is Multiplicative:

and:

Thus it is seen that:
 * $\sigma \left({r}\right) = \sigma \left({s}\right)$

Now we have:

and so it is seen that:
 * $r + s = \sigma \left({r}\right) = \sigma \left({s}\right)$

Hence the result, by definition of amicable numbers.