Euler's rule
Encyclopedia
Euler's rule, named after Leonhard Euler
, is a generalization of Thâbit ibn Kurrah rule
for finding amicable number
s. If a = 2m×(2n−m + 1) − 1, b = 2n×(2n−m + 1) − 1, and c = 2n+m×(2n−m + 1)2 − 1 are all prime
, for integers 0 < m < n, then 2n × a × b and 2n × c are amicable. This hypothesis is satisfied for the pairs (m,n) = (1,2), (3,4), (6,7), (1,8), and (29,40), but for no other pairs with n < 2500. The first three of these pairs yield the three pairs of amicable numbers discovered using the Thâbit ibn Kurrah rule.
The Euler's rule is that also that polyhedrons cannot be two polyhedrons connected by an edge or vertex
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...
, is a generalization of Thâbit ibn Kurrah rule
Thâbit 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....
for finding amicable number
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. If a = 2m×(2n−m + 1) − 1, b = 2n×(2n−m + 1) − 1, and c = 2n+m×(2n−m + 1)2 − 1 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...
, for integers 0 < m < n, then 2n × a × b and 2n × c are amicable. This hypothesis is satisfied for the pairs (m,n) = (1,2), (3,4), (6,7), (1,8), and (29,40), but for no other pairs with n < 2500. The first three of these pairs yield the three pairs of amicable numbers discovered using the Thâbit ibn Kurrah rule.
The Euler's rule is that also that polyhedrons cannot be two polyhedrons connected by an edge or vertex