Amicable Numbers

     Before this blog I had no idea what Amicable numbers were so I had to go out and learn about them and this is what I have learned. Amicable numbers are two integers greater than or equal to 1 whose sum of  proper positive divisors (any divisor of some positive integer n other than n) of one of the integers is equal to the other integer, and vice versa. A pair of amicable numbers form an aliquot sequence (a recursive sequence in which each term is the sum of the proper divisors of the previous term) of period 2. 

Amicable numbers were known by Pythagoreans who thought they had mystical properties. Amicable numbers are sometimes called friendly numbers, the story behind that is included on shyamsundergupta website in the amicable numbers section were it is said to be when Pythagoras was asked "what is a friend?" and replied that a friend is one "who is the other I, such as 220 and 284".  (220,284) are the smallest and most well know pair of amicable numbers. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, which if we sum together we get 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which if we sum together we get 220.
     A general formula has been derived from Iraqi mathematician Thābit ibn Qurra, the method for discovering amicable numbers states that if

p = 3 × 2^(n − 1 )− 1,

q = 3 × 2^n − 1,

r = 9 × 2^(2n − 1) − 1,

where n > 1 is an integer and p,q, and r are prime numbers, then 2n×p×q and 2n×r are a pair of amicable numbers. So if we choose are number to be 2 we get p = 3 x 2^((2) - 1) -1 = 5, q = 3 x 2^2 -1 = 11, and r = 9 x 2^(2(2)-1)-1 = 71. We can see that are p, q, and r are all prime numbers  and n > 1 so we have satisfied the hypothesis which means that 2n×p×q and 2n×r are a pair of amicable numbers so if we plug in our values we got for p, q, and r we get 2(2)×5×11 = 220 and 2(2)×(71) = 284 which as we know are amicable numbers. If you play around with this formula trying new n values like I did you will quickly realize it is not often you get p,q,and r to all be prime numbers. This method although is not perfect because it cannot generate co-prime amicable numbers but it is still unknown if a co-prime amicable number pair even exist.
     Euler also took an interest in amicable numbers, he found 59 new pairs of amicable numbers. Euler generalized Thābit's Rule in Euler's rule for amicable numbers. In Euler's rule it states let n be a positive number, and choose 0< x < n such that g = 2^(n-x) + 1. If
p = 2^x × g - 1
q = 2^n × g - 1
s = 2^(n+x) × g^2 - 1
are all prime numbers then
 (M,N) = (2^n × p × q, 2^n × s)
is an amicable pair. We can see that Euler made some slight modifications to Thābit's original conjecture and because of it was able to find new pairs of amicable numbers. Although Euler's modifications to the formula still could not fix the issue of the formula not being able to generate co-prime amicable numbers.

     Like a handful of topics in mathematics there are many things we have yet to figure out or prove. There are still many things we do not know about amicable numbers, for example in every know case, the pair of amicable numbers are either both even or both odd numbers. It is unknown if an even-odd pair of amicable numbers exist. I mentioned earlier that it is still not known if a pair of coprime amicable numbers exist since every known pair shares at least one common factor higher than 1. Using Thabit's formula or any similar formula (Euler's Rule) it is not possible to generate a pair of coprime amicable numbers. It will be interesting to see if someone finds a new method for calculating amicable numbers that is better than Euler's or Thabit's formula and if any of these special case pairs are ever discovered.