NUMBER

In Arithmetic, a number equal to the sum of all its divisors. Thus 6 is a perfect number, for its divisors are 1, 2, and 3; and 1 + 2 + 3 = 6. In like manner, 28 is a perfect numlier, for its divisors are 1, 2, 4, 7, 14; the sum of which = 28. In general, every number of the form 2"— ' (2n — 1), tho latter factor being a prime, is a perfect number, the sum el' its divisors being equal to the number itself. The difficulty, therefore, of finding perfect numbers arises from that of finding primes of the form 2" — 1, which is very laborious. The only values of x yet discovered which make 2" — la prime number are 2, 3, 5, 7, 13, 17, 19, and 31. Substituting these values of a succe ively iii the formula of the perfect number, we shall obtain the following results' If n = 2, then 2 (22 — 1) = 6 n = 3, » W — 1)=28 n = 5, 2< (* — 1) = 496 n = 7, 2« (27 — 1) = 8128 n = 13, 212(213 — 1) =33550330 n = 17, 216 (2n_i) = 8589869056 n = 19, 2is (2»—l) = 137438691328 7i = 31, 23«(23i — 1) =230584300813995212** all which are perfect numbers. [s. 931]