Let a(n) denote the sum of the positive divisors of the integer n and s(n) the sum of its aliquot parts, that is, of the divisors of n other than n itself; then,
Greek mathematicians had considered the problem of the determination of a perJect number (z~.~o~ do~6~), that is, a number n such that s(n) = n. EUCLID demonstrated that if 2 m --1 i s a prime, then 2m--1(2 m ---1)" is perfect. EuLER proved that any even perfect number must have this form, so that the question of the generality of EUCLID'S formula depends upon the existence of odd perfect numbers. This remains an open question today since no odd perfect number has been found but no demonstration of their non-existence has been found either.
The idea :of considering these perfect numbers goes back to the Pythagoreans. Such is also the case for the so-called amicable numbers (9~xo~ ~p~.o0; that is, pairs of numbers nl, n2 such that each is the sum of the aliquot parts of the other, i.e., s ( n l ) = n 2 and S(nz)~--nl --or a ( n ~) = a ( n 2 ) --~n l + n 2 . The Pythagoreans apparently only knew of the pair 220,284, and a metliod for finding further pairs (actually having the very restricted form h a = 2hP; n2 = 21'qr, with p; q, r different primes) was not discovered before the ninth century, by THWart" IroN QURRA. His rule was rediscovered in seventeenth century Europe, from which time research went further. Thus, EUL~R determined some sixty pairs, but not subject to the former restriction on their prime factorization.
Two further questions in relation to the sums of divisors arose during Islamic times:
--Find numbers nl, n2 such that s(nl) = s(n2).