Amicable numbers
Divisibility-based sets of integers | ||
|---|---|---|
| Overview |  | |
| Factorization forms | ||
| Constrained divisor sums | ||
| With many divisors | ||
| Aliquot sequence-related | ||
| Base-dependent | ||
| Other sets | ||
Amicable numbers are two different numbers so related that the sum of the proper divisors of one of the numbers is equal to the other. (A proper divisor of a number is a divisor other than the number itself.) A pair of amicable numbers constitutes an aliquot sequence of period 2. A related concept is that of a perfect number, which is a number which equals the sum of its own proper divisors, in other words a number which forms an aliquot sequence of period 1.
For example, the smallest pair of amicable numbers is (220, 284); for the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71, and 142, of which the sum is 220.
The first few amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368) (sequence A063990 in the OEIS).
History
Amicable numbers were known to the Pythagoreans, who credited them with many mystical properties. A general formula by which these numbers could be derived was invented circa 850 by 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'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. The second smallest pair, (1184, 1210), was discovered in 1866 by a then teen-aged B. Nicolò I. Paganini, having been overlooked by earlier mathematicians.
As of 1946 there were 390 known pairs, but the advent of computers has allowed the discovery of many thousands since then. Exhaustive searches have been carried out to find all pairs less than a given bound, this bound being extended from 108 in 1970, to 1010 in 1986, 1011 in 1993, and to a bound well over that today.
Rules for generating
Thābit's rule states that if
- p = 3 × 2n − 1 − 1,
- q = 3 × 2n − 1,
- r = 9 × 22n − 1 − 1,
where n > 1 is an integer and p, q, and r are prime numbers, then 2npq and 2nr are a pair of amicable numbers. This formula gives the pairs (220, 284) (n=2), (17296, 18416) (n=4), and (9363584, 9437056) (n=7), but no such pairs are known. Numbers of the form 3 × 2n − 1 are known as Thabit numbers. In order for Thābit's formula to produce an amicable pair, two consecutive Thabit numbers must be prime; this severely restricts the possible values of n.
A generalization of this is Euler's rule, which 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 2npq and 2nr are a pair of amicable numbers. Thābit's rule 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.
While these rules do generate some pairs of amicable numbers, many other pairs are known, so these rules are by no means comprehensive.
Regular pairs
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 prime to g and square free then the pair (m, n) is said to be regular, 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).
There no amicable pairs of type (1, j) for any j.
Other results
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. Also, every known pair shares at least one common 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 1067. Also, a pair of coprime amicable numbers cannot be generated by Thabit's formula (above), nor by any similar formula.
Notes
- ^ Costello, Patrick (2002-05-01). "New Amicable Pairs Of Type (2; 2) And Type (3; 2)" (PDF). MATHEMATICS OF COMPUTATION. 72 Number 241. American Mathematical Society: 489–497. Retrieved 2007-04-19.
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References
This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). Encyclopædia Britannica (11th ed.). Cambridge University Press. {{cite encyclopedia}}: Missing or empty |title= (help)
- Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers (pp. 145 - 147). London: Penguin Group.
- Weisstein, Eric W. "Amicable Pair". MathWorld.
- Weisstein, Eric W. "Thâbit ibn Kurrah Rule". MathWorld.
- Weisstein, Eric W. "Euler's Rule". MathWorld.