Amicable pairs Números amigos

1. Amicable pairs Números amigos

2. Números amigos Parejas de números (n, m) tq.: s(n) = s(m) Pythagoras said true friendship is comparable to the numbers 220 and 284 - this is the smallest amicable pair: s(220) = 284 and s(284) = 220. • 220 & 284: known to the Greeks • 1184 & 1210: discovered by Paganini at age 16 in 1866 • 17.163 & 18.416: discovered by Fermat in 1636 • 9.363.584 & 9.437.056: discovered by Descartes in 1638 • Fact: Over 1000 pairs of friendly numbers are now known! Pedersen counted more than 10.410.218 (on December, 29th, 2005) (Se desconoce si existen infinitos pares de amigos).

3. Pythagoras when asked, "What is a friend", replied that a friend is one "who is the other I" such as 220 and 284. The numbers 220 and 284 form the smallest pair of amicable numbers (also known as friendly numbers) known to Pythagoras.

4. It was not until 1636 that the great Pierre de Fermat discovered another pair of amicable numbers (17296, 18416). Later Descartes gave the third pair of amicable numbers i.e. (9363584, 9437056). In the 18th century great Euler drew up a list of 64 amiable pairs (two of which later shown to be unfriendly). B.N.I. Paganini, a 16 years Old Italian, startled the mathematical world in 1866 by announcing that the numbers 1184 and 1210 were friendly. It was the second lowest pair and had been completely overlooked until then, Even Eulers list of Amicable pairs does not contain it. Today(Oct,2005) about 10306909 pairs of amicable numbers is known.

5. Ibn Khaldum (1332-1406) sobre talismanes y números amigos. Siglo IX Abu-I-Hasan Thabit ibh Querra en Bagdad descubrió la primera fórmula para engendrar números amigos: • 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) 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 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.

6. 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.