## Tuesday, January 11, 2011

### AMICABLE NUMBERS

The pair of numbers 220 and 284 have the curious property that each "contains" the other. In what way? In the sense that the sum of the proper positive divisors of each, sum to the other.

The factors of 220 are: 1 2 4 5 10 11 20 22 44 55 110 220
The prime factors are: 2 * 2 * 5 * 11

The factors of 284 are: 1 2 4 71 142 284
The prime factors are: 2 * 2 * 71

For 220 : 1+2+4+5+10+11+20+22+44+55+110 = 284
For 284 : 1+2+4+71+142 = 220

Such pairs of numbers are called amicable numbers

List of Amicable numbers http://oeis.org/A063990

A Million New Amicable Pairs

There is no formula or method known to list all of the amicable numbers. But there are formulas for certain special types:

if n > 1 and each of p = 3*2^{n-1} - 1, q = 3*2^n - 1, and r = 9*2^{2n-1} - 1 are prime, then (2^n)*pq and (2^n)*r are amicable numbers.

>> It is not known whether there exist infinitely many pairs of amicable numbers.

>> There exist pairs of odd amicable numbers, like (12,285; 14,595) or (67,095; 71,145),
but it is not known whether there exists any pair with one of the numbers odd and one even.

>> It is not known whether there are pairs of relatively prime amicable numbers. However, H. J. Kanold proved that if there existed a pair of relatively prime amicable numbers, then each of the numbers had to be greater than 1023 and they would have together more than 20 prime factors.

How Euler Did It