Amicable Numbers and Euler

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 Two distinct positive integers a and b are said to form an amicable pair if the sum of the proper divisors of a is equal to b, and the sum of the proper divisors of b is equal to a.

A proper divisor of a number is a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1,2, and 3.



The ancient Greeks were already interested in amicable pairs, and they knew of an example: 220 and 284. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110. The proper divisors of 284 are 1, 2, 4, 71, and 142.

If we write s(n) for the sum of the proper divisors of n. We have



220 and 284 was the only amicable pair known to the ancient Greeks, but over the centuries, two more pairs were found: 17296 and 18416, and 9363584 and 9437056. Those three pairs were all that were known before Euler tackled the problem of finding amicable pairs, and he found 58 more pairs, thus increasing the supply of known pairs from 3 all the way up to 61. Not bad!

The Sum of Divisors Function

In order to say anything about amicable numbers and how to find them, it is helpful to know how to compute the sum of the divisors of a number. It turns out to be better to consider the sum of all divisors, rather than just the proper divisors, and we’ll turn the amicable number problem into a problem about the sum of all divisors momentarily.

Let n be a positive integer. We let σ(n) denote the sum of all the positive divisors of n.

Example.

There is a clean way to compute σ(n), at least if we know the prime factorization of n.

 Theorem: Suppose the prime factorization of n is 

Then 



Example. Since the prime factorizations of 220 and 284 are  

we have

Note the following consequence of the theorem

Let us now rephrase the amicable number problem in terms of the σ function. The sum of the proper divisors of a number n is simply s(n) = σ(n) − n: the sum of all the divisors, except for n itself. So, if a and b are amicable, that means that

or

So, finding amicable pairs is the same as finding pairs (a, b) with σ(a) = σ(b) = a + b

By 1946 there were 390 known pairs, but the advent of computers has allowed the discovery of many thousands since then. As of December 2020, there are over 1,22,6,469,501 known amicable pairs.

Can you find some other pairs of amicable numbers?
The genius of Euler percolates through all of mathematics. His accomplishments were truly extraordinary. 


References:
1) Amicable Numbers, Wikipedia

2) The mathematics of Euler, Simon Rubinstein-Salzedo, Euler Circle.


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