Two problems of number theory in Islamic times

Abstract

Let o{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, s(n) =■■ o{ri) n. Greek mathematicians had considered the problem of the determination of a perfect number (teXeloc dpt^oc), that is, a number n such that s(n) = n. Euclid demonstrated that if 2m 1 is a prime, then 2"I~I(2m 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 (ciXoi apiifyoi), that is, pairs of numbers at,, n2 such that each is the sum of the aliquot parts of the other, i.e., s(ni) = n2 and s(n2) = nA or (t(az,) v{n2) =:z nx + n2. The Pythagoreans apparently only knew of the pair 220,284, and a method for finding further pairs (actually having the very restricted form nx = 2hp, n2 2hqr, with /?, q, r different primes) was not discovered before the ninth century, by Thabit ibn Qurra. His rule was rediscovered in seventeenth century Europe, from which time research went further. Thus, Euler 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: