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:
“Off south, the bison multiply so fast
a slaughter’s mandatory every spring
and every spring the creeks get fat
and Kicking Horse fills up.”
—Richard Hugo (1923–1982)
“I remember a very important lesson that my father gave me when I was twelve or thirteen. He said, “You know, today I welded a perfect seam and I signed my name to it.” And I said, “But, Daddy, no one’s going to see it!” And he said, “Yeah, but I know it’s there.” So when I was working in kitchens, I did good work.”
—Toni Morrison (b. 1931)
“It is not the number of years we have behind us, but the number we have before us, that makes us careful and responsible and determined to find out the truth about everything.”
—George Bernard Shaw (1856–1950)