Amicable numbers
Two different natural numbers, each of which is a number mutually equal to the sum of the proper divisors of the other number, form a pair of friendly numbers.
Often referred to by the sum of the proper divisors of x. This allows the definition be formulated as:
Two integers a and b form a pair of friends numbers if: and.
The smallest number befriended pair is formed from the numbers 220 and 284. It is easy to check that the two numbers satisfy the definition:
In a friendly pair of numbers, the smaller number is always abundant and the greater number deficient.
- 3.1 Examples
- 4.1 Quasibefreundete numbers
- 4.2 Sociable numbers
Early mentions and the set of Thabit Ibn Qurra
For the first time mentioned Pythagoras about 500 BC the amicable numbers 220 and 284 on the question of what is a friend, he replied, " One who is another I like 220 and 284 "
1636 shared with Pierre de Fermat in a letter to Marin Mersenne that he had found the amicable numbers 17296 and 18416. However, Walter Borho determined in 2003 that this pair of numbers in the 14th century by Ibn al -Banna ( 1265-1321 ) was found as well as Kamaladdin Farist. It cites Ibn al -Banna with: "The numbers 17296 and 18416 are friends that Allah is an abundant, deficient other omniscient.. "
We used the set of Thabit Ibn Qurra:
The proof of this theorem can be found in the article about sums divider.
Examples
- For n = 2, x = 11, y = 5, z = 71, all primes. This results in
- For n = 3, z = 287 = 7 · 41 is not prime, ie, with n = 3 one finds no amicable numbers.
- For n = 4 the results found by Fermat befriended couple.
- For n = 7 calculated Descartes 1638 Friends 9,363,584 and 9,437,056. However, these were already known, according Borho in 1600, by Muhammad Baqir Yazdi.
Today it is known that one can determine no further amicable numbers for n ≤ 191600 with the set of Thabit.
A set of Leonhard Euler
Leonhard Euler generalized the theorem of Thabit:
For the special case k = 1 we obtain the set of Thabit.
1747 Euler found 30 more pairs of numbers befriended and published them in his De numeri amicabilibus. Three years later he published another 34 pairs of numbers, of which, however, two pairs were wrong.
Adrien -Marie Legendre in 1830 found another pair.
1866 showed the Italians as Niccolò Paganini I. (not the violin virtuoso ) as a 16- year-old, that in 1184 and 1210 numbers are befriended. This had been overlooked until then. It is the second smallest number befriended couple.
1946 Escott published the complete list of 233 friends pairs of numbers that were known until 1943.
1985 calculated Herman te Riele (Amsterdam ) all friendly numbers less than 10 billion - a total of 1427 pairs.
In 2007, nearly 12 million known befriended numbers.
It is believed that there are infinitely many numbers of friends, but a proof is not yet known.
The set of Walter Borho
More befriended numbers can be found by means of the set of Walter Borho:
Examples
A = 220 = 22 · 55 and B = 284 = 22 · 71 are friends. So are a = 4, u = 55 and s = 71, where s is prime. p = 127 is prime and not a divisor of a = 4
- N = 1: q1 = 56 · 127 to 1 · 547 = 7111 = 13 is not prime. For n = 1, therefore we obtain no new amicable numbers.
- N = 2: q1 and q2 = 903 223 = 65,032,127 are both prime. It follows: A1 = 220 · 1272 · 903 223 and B1 = 4 · 1272 · 65,032,127 are friendly numbers.
With the help of this theorem Borho found further 10,455 befriended numbers.
Related classes of numbers
Quasibefreundete numbers
In addition to the amicable numbers there is still a class of numbers which is similar to the amicable numbers: the quasibefreundeten numbers. They differ from the amicable numbers insofar as their divisors of the number itself, the 1 is not taken into account, ie only the non- trivial factors.
For example, the divider 48 has 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48 the number 75 has the splitter 1, 3, 5, 15, 25 and 75, the sum of the non-trivial divisor of 48, and the sum of the non-trivial divisor of 75.
The first quasibefreundeten number pairs are (48, 75 ), ( 140, 195), (1050, 1925) and (1575, 1648) (follow- A005276 in OEIS ).
Folksy numbers
If a chain ( finite sequence ) of more than two natural numbers before, each of which the sum of proper divisors of the predecessor and the first number is the sum of the proper divisors of the last number, it is called sociable numbers (English sociable numbers ).
- Example of a chain of order 5:
- Example of a chain of order 28:
Today ( March 2013 ) are 217 of these chains known: A list of aliquot cycles of length Greater Than 2 sub- aliquot sequences ( Content chains ) refers to those sequences in which the sum of proper divisors of a follower is equal to the following link. So the social numbers form periodic aliquot sequences.