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perfect number

A whole number that is equal to the sum of all its factors except itself. For example, 6 is a perfect number because its factors, 1, 2, and 3 add to give 6. The next smallest is 28 (the sum of 1 + 2 + 4 + 7 + 14). The ancient Christian scholar Augustine explained that God could have created the world in an instant but chose to do it in a perfect number of days, 6. Early Jewish commentators felt that the perfection of the universe was shown by the Moon's period of 28 days. The next in line are 496, 8128, and 33,550,336. As René Descartes pointed out: "Perfect numbers like perfect men are very rare." All end in 6 or 8, though what seems to be an alternating pattern of 6's and 8's for the first few perfect numbers doesn't continue. All are of the form 2n-1(2n-1), where 2n-1 is a Mersenne prime, so that the search for perfect numbers is the search for Mersenne primes. The largest one known, as of March 2003, is 4.27764198 × 108107891. It isn't known if there are infinitely many perfect numbers or if there are any odd perfect numbers.

A pseudoperfect number or semi-perfect number is a number equal to the sum of some of its divisors, e.g. 12 = 2 + 4 + 6, 20 = 1 + 4 + 5 + 10. An irreducible semi-perfect number is a semi-perfect number, none of whose factors is semi-perfect, e.g. 104. A quasi-perfect number would be a number n whose divisors (excluding itself) sum to n + 1, but it isn't known if such a number exists. A multiply perfect number is a number n whose divisors sum to a multiple of n. An example is 120, whose divisors (including itself) sum to 360 = 3 × 120. If the divisors sum to 3n, n is called multiply perfect of order 3, or tri-perfect. Ordinary perfect numbers are multiply perfect of order 2. Multiply perfect numbers are known of order up to 8.

Related entry

   • abundant number

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