Os primeiros 20 pares de números amigos

The pair 220 and 284 is the oldest named example of numerical kinship: the proper divisors of 220 add to 284, and those of 284 add back to 220. Iamblichus, writing in the fourth century CE, credited the discovery to Pythagoras and reported that the Pythagoreans treated the pair as an emblem of friendship. That attribution comes some eight hundred years after the fact and is better treated as tradition than as evidence; what is clear is that the pair was known in antiquity and that the association with friendship stuck to it.

The first substantial mathematics on the subject is Arabic. Thābit ibn Qurra, working in ninth-century Baghdad, proved a rule: if p = 3·2^(n−1) − 1, q = 3·2^n − 1 and r = 9·2^(2n−1) − 1 are all prime for some n > 1, then 2^n·p·q and 2^n·r are amicable. For n = 2 the rule returns 220 and 284; n = 4 gives 17296 and 18416; n = 7 gives 9363584 and 9437056. Those are the only values of n below 15 for which all three expressions come out prime, which is why the rule produces so few pairs despite being correct. Kamāl al-Dīn al-Fārisī rediscovered the n = 4 pair in the fourteenth century, and Muhammad Baqir Yazdi found the n = 7 pair in the seventeenth.

Europe arrived late and duplicated some of the work: Fermat announced 17296 and 18416 in 1636, Descartes announced 9363584 and 9437056 in 1638, and both were already known in the Islamic world. Leonhard Euler then changed the scale of the problem. He generalised Thābit's rule and published a list of thirty pairs in 1747, later extending it to sixty-four — two of which were eventually shown to be wrong.

Euler's methods all produced large pairs, and in doing so skipped the second-smallest one entirely. In 1866 a sixteen-year-old Italian, B. Nicolò I. Paganini — not the violinist, who had died a quarter-century earlier — pointed out that 1184 and 1210 are amicable. No new technique was involved. Nobody had checked.

Propriedades principais

• Two numbers are amicable when each equals the sum of the other’s proper divisors: 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284, and the divisors of 284 sum back to 220.
• For any amicable pair (m, n), σ(m) = σ(n) = m + n — for 220 and 284 that shared divisor sum is 504. It follows that the smaller member is always abundant and the larger always deficient.
• Thābit ibn Qurra’s rule: if 3·2^(n−1) − 1, 3·2^n − 1 and 9·2^(2n−1) − 1 are all prime, then 2^n times the product of the first two, and 2^n times the third, are amicable. For n below 15 this happens only at n = 2, 4 and 7.
• Counting pairs by their smaller member, there are 5 below ten thousand, 13 below one hundred thousand, 42 below one million and 108 below ten million.
• The smallest pair is 220 and 284; the smallest pair of odd numbers is 12285 and 14595.
• No amicable pair with one even and one odd member has ever been found, and no pair whose members are coprime is known either.
• Paul Erdős proved in 1955 that the amicable numbers have density zero — almost every integer belongs to no pair at all.
• Whether infinitely many amicable pairs exist is an open problem, even though computer searches have tabulated well over a billion of them.