Thabit ibn Qurra's Theorem

Thabit ibn Qurra's theorem, formulated by the ninth-century scholar Thabit ibn Qurra, gives a rule for constructing pairs of amicable numbers, pairs of integers each of which equals the sum of the other's proper divisors. The rule states that if p, q and r are prime numbers of the required form for some integer n greater than 1, then 2 to the n times p times q and 2 to the n times r form an amicable pair; it yields the known pairs 220 and 284 at n equal 2, 17296 and 18416 at n equal 4, and 9363584 and 9437056 at n equal 7, though no further pairs from the rule are currently known. The rule was rediscovered centuries later by Fermat in 1636 and Descartes in 1638, and Euler later generalized it into what is now called Euler's rule. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/