The pair of numbers 220 and 284 have the curious property that each "contains" the other. In what way? In the senseh that the sum of the proper positive divisors of each, sum to the other.
|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 (amicable means friendly--but there is a different set of number actually called friendly number).
Amicable numbers have a long history in magic and astrology, making love potions and talismans. As an example, some ancient Jewish commentators thought that Jacob gave his brother 220 sheep (200 female and 20 male) when he was afraid his brother was going to kill him (Genesis 32:14). The philosopher Iamblichus of Chalcis (ca. 250-330 A.D.) writes that the Pythagoreans knew of these numbers:
They call certain numbers amicable numbers, adopting virtues and social qualities to numbers, such as 284 and 220; for the parts of each have the power to generate the other.
Pythagoras is reported to have said that a friend is "one who is the other I, such as are 220 and 284." Now amicable numbers are most often (and most properly!) relegated to the exercise sections of elementary number theory texts.
There is no formula or method known to list all of the amicable numbers, but formulas for certain special types have been discovered throughout the years. Thabit ibn Kurrah (ca. A.D. 850) noted that
if n > 1 and each of p = 3.2n-1-1, q = 3.2n-1, and r = 9.22n-1-1 are prime, then 2npq and 2nr are amicable numbers.
It was centuries before this formula produced the second and third pair of amicable numbers! Fermat announced the pair 17,296 and 18,416 (n=4) in a letter to Mersenne in 1636. Descartes wrote to Mersenne in 1638 with the pair 9,363,584 and 9,437,056 (n=7). Euler then topped them both by adding a list of sixty-four new amicable pairs, however he made two errors. In 1909 one of his pairs was found to be not amicable, and in 1914 the same fate took a second pair. In 1866 a sixteen year old boy, Nicolo Paganini, discovered the pair (1184,1210) which was previously unknown.
Now extensive computer searches have found all such numbers with 10 or fewer digits and numerous larger examples, for a total of over 7500 amicable pairs. It is unknown if there are infinitely many pairs of amicable numbers. It is also unknown if there is a relatively prime pair of amicable numbers. If there is such a pair, they must be more than twenty-five digits long, and their product must be divisible by at least 22 distinct primes.
Related pages (outside of this work)
- Amicable pairs (an impressive attempt to list all known pairs)
- Amicable Numbers from Eric Weisstein's World of Mathematics
- Friendly pair of numbers from Eric Weisstein's World of Mathematics