Monday, September 28, 2015

What are Amicable Numbers?

Two numbers, k and j, make are amicable if the sum of the proper divisors of k is equal to j and if the sum of the proper divisors of j is equal to k.

History of the Amicable Numbers:

Although the equation that derives pairs of amicable numbers was popularized by Fermat and Descartes in the early 16th century, the concept and derivation of these unique numbers was first investigated by an Iranian mathematician named Thābit ibn Qurra (826, 901). Qurra was also one of the first mathematicians to describe the phenomenon that we now know as the Pythagorean Theorem. 

After Fermat and Descartes studied these unique numbers and rediscovered the pairs, Euler extended their work in the 18th century and discovered dozens of new pairs. After years of extension on these pairs of numbers, as of 2007, there are 12,000,000 known amicable pairs.

Interpretation of the Amicable Numbers:

Let's try to better our understanding of this definition by looking at an example. Firstly, the "proper divisors" of a number n are all of its divisors excluding n itself. Normally, we would list the divisors of the number 6 as

1, 2, 3, 6

However, the proper divisors of 6 (for this example we will denote this by p(6) ) would be listed as
p(6)= {1, 2, 3}

So, now that we know what a proper divisor is, we can continue interpreting the definition of an amicable pair.  Using our proper divisor notion, it follows that two numbers, k and j, are amicable if
Σp(k)=j and Σp(j)=k

The numbers 220 and 284 are  amicable, let's see why that is.

p(220)= {1, 2, 4, 71, 142}
p(284)= {1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110}

From the definition, it should follow that Σp(220)=284 and Σp(284)= 220, let's double check:

1 + 2 + 4 +71 + 142 = 284
1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 220.

So, we can see that 284 and 220 are amicable numbers.

Here is a great NumberPhile video for more explanantion on amicable numbers:

Finding Amicable Numbers:

So many great mathematicians studied amicable numbers because of the difficulty of finding these numbers. Part of this difficulty lies within the relationship between amicable numbers and prime numbers. This can be seen in Euler's rule for finding amicable numbers as follows:

"Euler's rule is a generalization of the Thâbit ibn Qurra theorem. It states that if

p = (2(n - m)+1) × 2m − 1,
q = (2(n - m)+1) × 2n − 1,
r = (2(n - m)+1)2 × 2m + n − 1,
where n > m > 0 are integers and p, q, and r are prime numbers, then 2n×p×q and 2n×r are a pair of amicable numbers."

Where Do We See Amicable Numbers?:

 Although there is no direct application of these pairs of integers, amicable numbers have a significant historical impact. Not only did the search for these pairs attract so many recognizable mathematicians over time, but they were representations of "friendship." In Genesis 32:14, Jacob gifted Esau with 220 goats, a number chosen to exemplify his love. 

Another instance in which amicable pairs were used to portray positive relationships was in Medieval horoscope projections. If two people were given a horoscope containing the numbers 220 and 284, then this indicated a future romance between them. These whimsical applications of amicable numbers further emphasizes the natural attraction that people have for these pairs.


  1. I like how you explain the difference between proper divisors and divisors, and show step by step what it means to be an amicable number for 220 and 284. After watching the video, a possible extension maybe why a pair of amicable numbers wasn't found until much later.

  2. Great mathematical exposition, good connections and resources. All it needs to be an exemplar is consolidation. I often recommend a what, so what or now what. This seems like a good 'so what' spot.

    Other Cs: +