Al-Baghdadi’s Method for Finding Equivalent Numbers
Al-Baghdadi gives a very elegant method for finding Equivalent numbers. The method is best illustrated with an example. Suppose one is attempting to find equivalent numbers n and m such that their proper divisors add up to, say, 23. In mathematical terms, given a positive integer k, we attempt to find n and m such that σo(n) = oo(m) = k. Below is the algorithm for finding equivalent numbers:
1. Subtract 23 from 1. Next, find two distinct prime numbers such that, when summed, they equal 22.
5 2. These two pairs are (3, 19) and (5, 17).
3. Next, multiply out each set separately. So, 3 19 = 57 and 5.17 = 85
4. The resulting numbers, 85 and 57, are then equivalent numbers.
One can follow this method for any positive integer k. Al-Baghdadi, however, stipulates two contingencies to this method: “But, whenever one subtracts from the [given] number 1 and the remainder is in no way divisible into two unidentical primes, or is so only one way, the said rule will fail” [21].
Discoveries in Number Theory
The discipline of Number Theory relies heavily on advancements in calculus, abstract algebra, and formal logic. Thus, number theory was weakly studied during the time of the Islamic civilizations. Only historically recently has number theory burgeoned due to advancements in calculus and abstract algebra, yet it is quite astonishing to learn that, even before such highly advanced discoveries and methods, two significant results in number theory were discovered, only using premodern algebra
Aal-e-Qutub Aal-e-Syed Abdullah Shah Ghazi 