a formula for amicable pairsAFormulaForAmicablePairs2006-04-27 16:27:562006-04-28 09:31:07Definitionamicable numbers% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Cl}{\operatorname{Cl}}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.
\begin{thm}
Let $n\geq 1$ be a natural number and suppose that the numbers
$$3\cdot 2^n -1, \quad 3\cdot 2^{n-1}-1 \quad \text{ and } \quad 9\cdot 2^{2n-1}-1$$
are all prime. Then the numbers:
$$2^n(3\cdot 2^n-1)(3\cdot 2^{n-1}-1)\quad \text{ and } \quad 2^n(9\cdot 2^{2n-1}-1)$$
are amicable numbers.
\end{thm}
\begin{exa}
When $n=2$ one has:
$$3\cdot 2^2 -1=11, \quad 3\cdot 2^{2-1}-1=5 \quad \text{ and } \quad 9\cdot 2^{4-1}-1=71$$
which are all primes. Thus, the numbers:
$$2^2(3\cdot 2^2-1)(3\cdot 2^{2-1}-1)=220\quad \text{ and } \quad 2^2(9\cdot 2^{4-1}-1)=284$$
form an amicable pair. In fact, this is the smallest amicable pair. For $n=4$ one obtains the amicable pair $17296$ and $18416$.
\end{exa}