In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.
For a given natural number k, a number n is called k-perfect (or k-fold perfect) if and only if the sum of all positive divisors of n (the divisor function, σ(n)) is equal to kn; a number is thus perfect if and only if it is 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number. As of July 2004, k-perfect numbers are known for each value of k up to 11.
It can be proven that:
- For a given prime number p, if n is p-perfect and p does not divide n, then pn is (p+1)-perfect. This implies that an integer n is a 3-perfect number divisible by 2 but not by 4, if and only if n/2 is an odd perfect number, of which none are known.
- If 3n is 4k-perfect and 3 does not divide n, then n is 3k-perfect.
Read more about Multiply Perfect Number: Smallest k-perfect Numbers
Famous quotes containing the words multiply, perfect and/or number:
“We go on multiplying our conveniences only to multiply our cares. We increase our possessions only to the enlargement of our anxieties.”
—Anna C. Brackett (1836–1911)
“A single thankful thought towards heaven is the most perfect of all prayers.”
—Gotthold Ephraim Lessing (1729–1881)
“Cole’s Hill was the scene of the secret night burials of those who died during the first year of the settlement. Corn was planted over their graves so that the Indians should not know how many of their number had perished.”
—For the State of Massachusetts, U.S. public relief program (1935-1943)