Thabit's Rule

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

are all prime.

Then:
 * $\tuple {2^n a b, 2^n c}$

forms an amicable pair.

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

Let $\map {\sigma_1} k$ denote the divisor sum of an integer $k$.

From Divisor Sum of Power of 2:
 * $\map {\sigma_1} {2^n} = 2^{n + 1} - 1$

From Divisor Sum of Prime Number:

From Divisor Sum Function is Multiplicative:

and:

Thus it is seen that:
 * $\map {\sigma_1} r = \map {\sigma_1} s$

Now we have:

and so it is seen that:
 * $r + s = \map {\sigma_1} r = \map {\sigma_1} s$

Hence the result, by definition of amicable pair.

Also see

 * Definition:Thabit Pair