As we learned in class, a pair of integers (m, n) are called an amicable pair if the sum of the positive divisors of n is equal to m and the sum of all the positive divisors of m is equal to n.

Since the time of Pythagoras, 220 and 284 have been observed to be the smallest amicable pair. The next smallest pair is 1184 and 1210, and was discovered by Nicolo Paganini, a 16-year-old schoolboy in 1866. (What a smartie!!)

Thabit ibn Qurra, an Arab mathematician, physician, and astronomer wrote a book titled *Book on the Determination of Amicable Numbers.* In this book, he describes the rule Euclid developed for perfect numbers and a formula for obtaining amicable pairs. You need to find a number n that is greater than 1 that makes a, b, AND c prime:

a = 3 × 2^n - 1

b = 3 × 2^n-1 - 1

c = 9 × 2^2n-1 - 1

The two numbers 2^n × a × b and 2^n × c will be amicable and are called Thabit pairs. So far, the only values that make all three numbers prime are n = 2, 4, or 7. are the only values found so far that make all three numbers prime. Sadly, Thabit's rule does not give all amicable pairs, but is merely a pattern he discovered. There isn't a rule for finding amicable pairs.