Amicable numbers

Amicable numbers were known to the Pythagoreans, who credited them with many mystical properties. A general formula by which some of these numbers could be derived was invented circa 850 by the Iraqi mathematician Thābit ibn Qurra (826–901). Other Arab mathematicians who studied amicable numbers are al-Majriti (died 1007), al-Baghdadi (980–1037), and al-Fārisī (1260–1320). The Iranian mathematician Muhammad Baqir Yazdi (16th century) discovered the pair (9363584, 9437056), though this has often been attributed to Descartes.^[1] Much of the work of Eastern mathematicians in this area has been forgotten.

Thābit ibn Qurra's formula was rediscovered by Fermat (1601–1665) and Descartes (1596–1650), to whom it is sometimes ascribed, and extended by Euler (1707–1783). It was extended further by Borho in 1972. Fermat and Descartes also rediscovered pairs of amicable numbers known to Arab mathematicians. Euler also discovered dozens of new pairs.^[2] The second smallest pair, (1184, 1210), was discovered in 1867 by 16-year-old B. Nicolò I. Paganini (not to be confused with the composer and violinist), having been overlooked by earlier mathematicians.^[3][4]

There are over 1 billion known amicable pairs.^[5]

While these rules do generate some pairs of amicable numbers, many other pairs are known, so these rules are by no means comprehensive.

In particular, the two rules below produce only even amicable pairs, so they are of no interest for the open problem of finding amicable pairs coprime to 210 = 2·3·5·7, while over 1000 pairs coprime to 30 = 2·3·5 are known [García, Pedersen & te Riele (2003), Sándor & Crstici (2004)].

The Thābit ibn Qurrah theorem is a method for discovering amicable numbers invented in the 9th century by the Arab mathematician Thābit ibn Qurrah.^[6]

It states that if $${\displaystyle {\begin{aligned}p&=3\times 2^{n-1}-1,\\q&=3\times 2^{n}-1,\\r&=9\times 2^{2n-1}-1,\end{aligned}}}$$

where n > 1 is an integer and p, q, r are prime numbers, then 2ⁿ × p × q and 2ⁿ × r are a pair of amicable numbers. This formula gives the pairs (220, 284) for n = 2, (17296, 18416) for n = 4, and (9363584, 9437056) for n = 7, but no other such pairs are known. Numbers of the form 3 × 2ⁿ − 1 are known as Thabit numbers. In order for Ibn Qurrah's formula to produce an amicable pair, two consecutive Thabit numbers must be prime; this severely restricts the possible values of n.

To establish the theorem, Thâbit ibn Qurra proved nine lemmas divided into two groups. The first three lemmas deal with the determination of the aliquot parts of a natural integer. The second group of lemmas deals more specifically with the formation of perfect, abundant and deficient numbers.^[6]

Euler's rule is a generalization of the Thâbit ibn Qurra theorem. It states that if $${\displaystyle {\begin{aligned}p&=(2^{n-m}+1)\times 2^{m}-1,\\q&=(2^{n-m}+1)\times 2^{n}-1,\\r&=(2^{n-m}+1)^{2}\times 2^{m+n}-1,\end{aligned}}}$$ where n > m > 0 are integers and p, q, r are prime numbers, then 2ⁿ × p × q and 2ⁿ × r are a pair of amicable numbers.

Note that $${\displaystyle {\begin{aligned}pq&=r-(2^{n-m}+1)(2^{n}+2^{m})+2\\pq&=r-((2^{n-m}+1)(2^{n}+2^{m})-2)\\pq&=r-(p+q)\end{aligned}}}$$ thus ${\displaystyle pq+(p+q)=r}$

Thābit ibn Qurra's theorem corresponds to the case m = n − 1. Euler's rule creates additional amicable pairs for (m,n) = (1,8), (29,40) with no others being known. Euler (1747 & 1750) overall found 58 new pairs increasing the number of pairs that were then known to 61.^[2][7]

Let (m, n) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair (m, n) is said to be regular (sequence A215491 in the OEIS); otherwise, it is called irregular or exotic. If (m, n) is regular and M and N have i and j prime factors respectively, then (m, n) is said to be of type (i, j).

For example, with (m, n) = (220, 284), the greatest common divisor is 4 and so M = 55 and N = 71. Therefore, (220, 284) is regular of type (2, 1).

An amicable pair (m, n) is twin if there are no integers between m and n belonging to any other amicable pair (sequence A273259 in the OEIS).

In every known case, the numbers of a pair are either both even or both odd. It is not known whether an even-odd pair of amicable numbers exists, but if it does, the even number must either be a square number or twice one, and the odd number must be a square number. However, amicable numbers where the two members have different smallest prime factors do exist: there are seven such pairs known.^[8] Also, every known pair shares at least one common prime factor. It is not known whether a pair of coprime amicable numbers exists, though if any does, the product of the two must be greater than 10⁶⁵.^[9][10] Also, a pair of co-prime amicable numbers cannot be generated by Thabit's formula (above), nor by any similar formula.

In 1955 Paul Erdős showed that the density of amicable numbers, relative to the positive integers, was 0.^[11]

In 1968 Martin Gardner noted that most even amicable pairs have sums divisible by 9,^[12] and that a rule for characterizing the exceptions (sequence A291550 in the OEIS) was obtained.^[13]

According to the sum of amicable pairs conjecture, as the number of the amicable numbers approaches infinity, the percentage of the sums of the amicable pairs divisible by ten approaches 100% (sequence A291422 in the OEIS).

Although all amicable pairs up to 10,000 are even pairs, the proportion of odd amicable pairs increases steadily towards higher numbers, and presumably there are more of them than of the even amicable pairs (sequence A360054 in OEIS).

There are amicable pairs where the sum of one number from the first pair and one number from the second pair equals the sum of the remaining two numbers, e.g. 67212 = 220 + 66992 = 284 + 66928 where (220, 284) and (66928, 66992) are two amicable pairs (sequence A359334 in OEIS).

Gaussian integer amicable pairs exist,^[14][15] e.g. s(8008 + 3960i) = 4232 − 8280i and s(4232 − 8280i) = 8008 + 3960i.^[16]

Amicable numbers ${\displaystyle (m,n)}$ satisfy ${\displaystyle \sigma (m)-m=n}$ and ${\displaystyle \sigma (n)-n=m}$ which can be written together as ${\displaystyle \sigma (m)=\sigma (n)=m+n}$. This can be generalized to larger tuples, say ${\displaystyle (n_{1},n_{2},\ldots ,n_{k})}$, where we require

${\displaystyle \sigma (n_{1})=\sigma (n_{2})=\dots =\sigma (n_{k})=n_{1}+n_{2}+\dots +n_{k}}$

For example, (1980, 2016, 2556) is an amicable triple (sequence A125490 in the OEIS), and (3270960, 3361680, 3461040, 3834000) is an amicable quadruple (sequence A036471 in the OEIS).

Amicable multisets are defined analogously and generalizes this a bit further (sequence A259307 in the OEIS).

Sociable numbers are the numbers in cyclic lists of numbers (with a length greater than 2) where each number is the sum of the proper divisors of the preceding number. For example, ${\displaystyle 1264460\mapsto 1547860\mapsto 1727636\mapsto 1305184\mapsto 1264460\mapsto \dots }$ are sociable numbers of order 4.

The aliquot sequence can be represented as a directed graph, ${\displaystyle G_{n,s}}$, for a given integer ${\displaystyle n}$, where ${\displaystyle s(k)}$ denotes the sum of the proper divisors of ${\displaystyle k}$.^[17] Cycles in ${\displaystyle G_{n,s}}$ represent sociable numbers within the interval ${\displaystyle [1,n]}$. Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs.

• Betrothed numbers (quasi-amicable numbers)
• Amicable triple - Three-number variation of Amicable numbers.