a formula for amicable pairs

The following formula is due to Thabit ibn Qurra (836-901), a mathematician who worked in Baghdad’s “House of Wisdom” translating Greek and Syrian works (such as Apollonius’s “Conics” or works of Euclid and Archimedes). As he translated the texts, ibn Qurra produced a mathematical body of his own.

Theorem.

Let $n\mathrm{\ge }\mathrm{1}$ be a natural number and suppose that the numbers

$$3\cdot 2^n-1,3\cdot 2^{n-1}-1\mathit{\hspace{1em}}\mathit{\text{ and }}\mathit{\hspace{1em}}9\cdot 2^{2n-1}-1$$

are all prime. Then the numbers:

$$2^n(3\cdot 2^n-1)(3\cdot 2^{n-1}-1)\mathit{\hspace{1em}}\mathit{\text{ and }}\mathit{\hspace{1em}}{2}^n(9\cdot 2^{2n-1}-1)$$

are amicable numbers.

Example.

When $n=2$ one has:

$$3\cdot 2^2-1=11,3\cdot 2^{2-1}-1=5\mathit{\hspace{1em}}\text{ and }\mathit{\hspace{1em}}9\cdot 2^{4-1}-1=71$$

which are all primes. Thus, the numbers:

$$2^2(3\cdot 2^2-1)(3\cdot 2^{2-1}-1)=220\mathit{\hspace{1em}}\text{ and }\mathit{\hspace{1em}}{2}^2(9\cdot 2^{4-1}-1)=284$$

form an amicable pair. In fact, this is the smallest amicable pair. For $n=4$ one obtains the amicable pair $17296$ and $18416$.