Amicable numbers

Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. Formally, an amicable pair $(m, n)$ satisfies the condition $\sigma(m) - m = n$ and $\sigma(n) - n = m$, where $\sigma(x)$ denotes the sum of all positive divisors of $x$, including $x$ itself. Equivalently, the sum of the proper divisors (divisors excluding the number itself) of each number equals the other. The smallest and most famous amicable pair is (220, 284).

Definition and Examples

Let $s(n)$ denote the aliquot sum of $n$, which is the sum of all proper divisors of $n$. Two distinct positive integers $m$ and $n$ form an amicable pair if $s(m) = n$ and $s(n) = m$.

The pair (220, 284) is the smallest amicable pair. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, which sum to 284. The proper divisors of 284 are 1, 2, 4, 71, and 142, which sum to 220. Other small amicable pairs include (1184, 1210), (2620, 2924), (5020, 5564), and (6232, 6368).

History

The concept of amicable numbers was first recognized by the Pythagoreans, who attributed mystical and philosophical significance to the pair (220, 284). In the 9th century, the Arab mathematician Thabit ibn Qurra discovered a general formula for generating amicable numbers, which yielded the pairs (220, 284), (17296, 18416), and (9363584, 9437056).

In the 17th century, René Descartes and Pierre de Fermat independently rediscovered Thabit's rule and found new pairs. Leonhard Euler significantly expanded the known list in the 18th century, discovering dozens of new amicable pairs and generalizing Thabit's formula. In 1866, a 16-year-old Italian mathematician named Nicolò Paganini discovered the pair (1184, 1210), which had been overlooked by earlier mathematicians because it is the second smallest pair.

With the advent of modern computers, the search for amicable numbers has been vastly accelerated. As of recent computational efforts, over a billion amicable pairs have been discovered, with the vast majority found through distributed computing projects.

Generation Rules

While there is no single formula that generates all amicable pairs, several mathematical rules have been developed to produce specific families of pairs.

Thabit ibn Qurra's Theorem

Thabit ibn Qurra's theorem states that if $p = 3 \cdot 2^{n-1} - 1$, $q = 3 \cdot 2^n - 1$, and $r = 9 \cdot 2^{2n-1} - 1$ are all prime numbers for some integer $n > 1$, then $2^n \cdot p \cdot q$ and $2^n \cdot r$ form an amicable pair. This formula successfully generates the pairs for $n = 2, 4,$ and $7$, but it does not produce any other amicable pairs for higher values of $n$ within computationally verified ranges.

Euler's Rule

Leonhard Euler generalized Thabit's theorem. Euler's rule states that if $p = (2^{n-m}+1) \cdot 2^m - 1$, $q = (2^{n-m}+1) \cdot 2^n - 1$, and $r = (2^{n-m}+1)^2 \cdot 2^{m+n} - 1$ are all prime numbers for integers $n > m > 0$, then $2^{m+n} \cdot p \cdot q$ and $2^{m+n} \cdot r$ form an amicable pair. Thabit's rule is a special case of Euler's rule where $m = n - 1$. Euler's rule has been used to find several additional pairs, though it still does not account for all known amicable pairs.

Properties and Unsolved Problems

Amicable numbers exhibit several interesting properties and are associated with prominent unsolved problems in number theory.

One of the most notable open questions is whether an amicable pair can consist of one even and one odd number. All known amicable pairs share the same parity; they are either both even or both odd. If an odd-even amicable pair exists, the even number must be a perfect square or twice a perfect square, and the odd number must be a perfect square. Furthermore, the product of the two numbers must be extraordinarily large, making it computationally infeasible to find with current technology.

Another unsolved problem is whether there exists a pair of coprime amicable numbers. It has been proven that if a coprime amicable pair exists, their product must be exceedingly large, and they cannot be generated by Thabit's or Euler's rules.

Additionally, it is unknown whether there are infinitely many amicable pairs. While heuristics and computational evidence strongly suggest that the number of amicable pairs is infinite, a rigorous mathematical proof remains elusive.

Generalizations

The concept of amicable numbers can be generalized in several directions within number theory.

A perfect number is a number that is its own amicable partner, meaning the sum of its proper divisors equals the number itself. Amicable numbers can be viewed as a cycle of length two in the aliquot sequence. This leads to the concept of sociable numbers, which form cycles of length greater than two. For example, a sociable cycle of length 4 consists of four numbers where the sum of the proper divisors of each number equals the next number in the cycle, and the sum of the proper divisors of the last number equals the first.

Amicable tuples extend the concept to more than two numbers. Furthermore, the definition can be adapted to other number systems, such as Gaussian integers, leading to the study of complex amicable numbers. The study of these generalizations continues to be an active area of research in recreational and computational number theory.