Amicable number
Encyclopedia
Amicable numbers are two different number
s so related that the sum
of the proper divisors of each is equal to the other number. (A proper divisor of a number is a positive integer divisor other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3.) 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. Numbers that are members of an aliquot sequence with period greater than 2 are known as sociable number
s.
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) .
, who credited them with many mystical properties. A general formula by which some of 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 Iran
ian mathematician Muhammad Baqir Yazdi
(16th century) discovered the pair (9363584, 9437056), though this has often been attributed to Descartes
. 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 teenage 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.
In 2007, there were almost 12,000,000 known amicable pairs.
states that if
where n > 1 is an integer
and p, q, and r are prime number
s, then 2npq and 2nr 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 number
s. 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
where n>m> 0 are integer
s and p, q, and r are prime number
s, 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.
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, 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).
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. Also, every known pair shares at least one common factor
, higher than 1. 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.
In 1955, Paul Erdős
showed that the density of amicable numbers, relative to the positive integers, was 0.
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s so related that the sum
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
of the proper divisors of each is equal to the other number. (A proper divisor of a number is a positive integer divisor other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3.) A pair of amicable numbers constitutes an aliquot sequence
Aliquot sequence
In mathematics, an aliquot sequence is a recursive sequence in which each term is the sum of the proper divisors of the previous term. The aliquot sequence starting with a positive integer k can be defined formally in terms of the sum-of-divisors function σ1 in the following way:For example, the...
of period
Periodic sequence
In mathematics, a periodic sequence is a sequence for which the same terms are repeated over and over:The number p of repeated terms is called the period.-Definition:A periodic sequence is a sequence a1, a2, a3, ... satisfying...
2. A related concept is that of a perfect number
Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...
, 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. Numbers that are members of an aliquot sequence with period greater than 2 are known as sociable number
Sociable number
Sociable numbers are generalizations of the concepts of amicable numbers and perfect numbers. A set of sociable numbers is a kind of aliquot sequence, or a sequence of numbers each of whose numbers is the sum of the factors of the preceding number, excluding the preceding number itself...
s.
For example, the smallest pair of amicable numbers is (220
220 (number)
220 is the natural number following 219 and preceding 221.-In mathematics:It is a composite number, with its divisors being 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, making it an amicable number with 284...
, 284
284 (number)
Two hundred eighty-four is the natural number following 283 and preceding 285.Its divisors are 1, 2, 4, 71, and 142, adding up to 220, in turn, the divisors of 220 add up to 284, making the two a pair of amicable numbers....
); 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) .
History
Amicable numbers were known to the PythagoreansPythagoreanism
Pythagoreanism was the system of esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were considerably influenced by mathematics. Pythagoreanism originated in the 5th century BCE and greatly influenced Platonism...
, who credited them with many mystical properties. A general formula by which some of these numbers could be derived was invented circa 850 by Thābit ibn Qurra
Thabit ibn Qurra
' was a mathematician, physician, astronomer and translator of the Islamic Golden Age.Ibn Qurra made important discoveries in algebra, geometry and astronomy...
(826-901). Other Arab mathematicians who studied amicable numbers are al-Majriti
Maslamah Ibn Ahmad al-Majriti
Maslama al-Majriti or Abu al-Qasim al-Qurtubi al-Majriti was a Muslim astronomer, chemist, mathematician, economist and Scholar in Islamic Spain...
(died 1007), al-Baghdadi
Ibn Tahir al-Baghdadi
Abu Mansur Abd al-Qahir ibn Tahir ibn Muhammad ibn Abdallah al-Tamimi al-Shaffi al-Baghdadi was an Arabian mathematician from Baghdad who is best known for his treatise al-Takmila fi'l-Hisab. It contains results in number theory, and comments on works by al-Khwarizmi which are now lost.-...
(980-1037), and al-Fārisī (1260–1320). The Iran
Iran
Iran , officially the Islamic Republic of Iran , is a country in Southern and Western Asia. The name "Iran" has been in use natively since the Sassanian era and came into use internationally in 1935, before which the country was known to the Western world as Persia...
ian mathematician Muhammad Baqir Yazdi
Muhammad Baqir Yazdi
Muhammad Baqir Yazdi was an Iranian mathematician who lived in the 16th century. He gave the pair of amicable numbers 9,363,584 and 9,437,056 many years before Euler's contribution to amicable numbers....
(16th century) discovered the pair (9363584, 9437056), though this has often been attributed to Descartes
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...
. Much of the work of Eastern mathematicians in this area has been forgotten.
Thābit's formula was rediscovered by Fermat
Pierre de Fermat
Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...
(1601–1665) and Descartes
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...
(1596–1650), to whom it is sometimes ascribed, and extended by Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
(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 teenage 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.
In 2007, there were almost 12,000,000 known amicable pairs.
Rules for generation
Thābit's ruleThâbit ibn Kurrah rule
Thâbit ibn Kurrah rule is a method for discovering amicable numbers invented in the tenth century by the Arab mathematician Thâbit ibn Kurrah. A later generalization of this rule is Euler'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
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
and p, q, and r are prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s, then 2npq and 2nr 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 number
Thabit number
In number theory, a Thabit number, Thâbit ibn Kurrah number, or 321 number is an integer of the form 3·2n−1 for a non-negative integer n...
s. 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
Euler's rule
Euler's rule, named after Leonhard Euler, is a generalization of Thâbit ibn Kurrah rule for finding amicable numbers. If a = 2m× − 1, b = 2n× − 1, and c = 2n+m×2 − 1 are all prime, for integers 0 Euler's rule, named after Leonhard Euler, is a generalization of Thâbit ibn Kurrah rule for finding...
, 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 integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s and p, q, and r are prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s, 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 divisorGreatest common divisor
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
of m and n. If M and N are both coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
to g and square free
Square-free integer
In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32...
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).
Other results
In every known case, the numbers of a pair are either both evenEven and odd numbers
In mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2...
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. Also, every known pair shares at least one common factor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
, higher than 1. It is not known whether a pair of coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
amicable numbers exists, though if any does, the product
Product (mathematics)
In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication...
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.
In 1955, Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
showed that the density of amicable numbers, relative to the positive integers, was 0.