Number theory is a branch of math that explores the patterns and nature of positive whole numbers. We begin with introducing a concept termed amicable numbers, followed by a proof on finding amicable numbers, then turn to equivalent numbers, followed by an algorithm to obtain equivalent numbers.

4.1 Introduction to Amicable Numbers

The only application or use for these numbers is the original one – you insert a pair of amicable numbers into a pair of amulets, of which you wear one yourself and give the other to your beloved!

-John Conway

In the following sections, we build up to a complex proof for finding amicable numbers. Amicable (friendly) numbers are just as they sound! They are two numbers, say x and y, such that the sum of the divisors of x (except for x itself) will give you y, and the sum of the divisors of y (except for y itself) will give you x. So, for example, let x 220 let y = 284. The proper divisors of x are 1, 2, 5, 10, and 11, 20, 22, 44, 55, and 110. The proper divisors of y are 1, 2, 4, 71, and 142. So, =

and

1+2+5+10 +11+20 +22+44 +55 + 110 = 284

1+2+4+71 + 142 = 220.

Thus, 220 and 284 are amicable numbers.

We next introduce Thabit Ibn Qurra, who developed a rigorous proof for finding amicable numbers.

4.2 Thabit Ibn Qurra

Thabit Ibn Qurra was born in 836 CE during the Abbasid Caliphate in Harran, Upper Mesopotamia. Ibn Qurra grew up a Sabian, but later converted to Islam [14]. Ibn Qurra was persuaded by one of the three Banu-Musa brothers, who were all prolific mathematicians at the time, to go to Baghdad and learn from the Banu Musa brothers. Ibn Qurra was fluent in Syriac, Greek, and Arabic, and thus translated many works of Greek origin to Arabic. and he made discoveries in algebra, geometry, astronomy, physics, and number theory. He estimated the sidereal year as 365 days, 6 hours, 9 minutes, and 12 seconds. He overestimated the sidereal year by 3.5 seconds.