Thâbit ibn Kurrah rule
Encyclopedia
Thâbit ibn Kurrah rule is a method for discovering amicable number
s invented in the tenth century by the Arab
mathematician
Thâbit ibn Kurrah. A later generalization of this rule is Euler's rule
.
The rule is given in terms of Thâbit ibn Kurrah numbers. For any natural number
n, the nth Thâbit ibn Kurrah number is Kn = 3×2n − 1. The first ten Thâbit ibn Kurrah numbers are 2, 5, 11, 23, 47, 95, 191, 383, 767, and 1535.
Thâbit ibn Kurrah showed that if Kn, Kn−1, and 3×K2n−1 + 2 are all prime
, then the pair (2n×Kn×Kn−1, 2n×(3×K2n−1 + 2)) is amicable.
The hypothesis is met in only three cases, n = 2, 4, and 7, giving amicable pairs (220, 284), (17296, 18416), and (9363584, 9437056).
Reference:
Amicable number
Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. A pair of amicable numbers constitutes an aliquot sequence of period 2...
s invented in the tenth century by the Arab
Arab
Arab people, also known as Arabs , are a panethnicity primarily living in the Arab world, which is located in Western Asia and North Africa. They are identified as such on one or more of genealogical, linguistic, or cultural grounds, with tribal affiliations, and intra-tribal relationships playing...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Thâbit ibn Kurrah. A later generalization of this rule 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...
.
The rule is given in terms of Thâbit ibn Kurrah numbers. For any natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
n, the nth Thâbit ibn Kurrah number is Kn = 3×2n − 1. The first ten Thâbit ibn Kurrah numbers are 2, 5, 11, 23, 47, 95, 191, 383, 767, and 1535.
Thâbit ibn Kurrah showed that if Kn, Kn−1, and 3×K2n−1 + 2 are all prime
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...
, then the pair (2n×Kn×Kn−1, 2n×(3×K2n−1 + 2)) is amicable.
The hypothesis is met in only three cases, n = 2, 4, and 7, giving amicable pairs (220, 284), (17296, 18416), and (9363584, 9437056).
Reference: