Recall that a perfect number is an integer that is the sum
of its aliquot divisors, that is, all of its positive
divisors except itself. Another way to say this is:
n is perfect if the sum of all of its positive
divisors, denoted sigma(n), is twice n. Any
positive integer n which divides the sum of its
positive divisors is called multiply perfect or
k-perfect where k is the index
sigma(n)/n. For example, here are the
smallest multiply perfect numbers for their index:
||4-perfect||Descartes, c. 1638|
|5||14182439040||5-perfect||Descartes, c. 1638|
Fermat (not Carmichael) was the first to find a 6-perfect number (in 1643):
You might want to try your hand at proving the following theorems:
- If n is 3-perfect and 3 does not divide n, then 3n is 4-perfect.
- If n is 5-perfect and 5 does not divide n, then 5n is 6-perfect.
- If general, suppose p is prime. If n
is p-perfect and p does not divide n,
then pn is (p+1)-perfect.
- If 3n is 4k-perfect and 3 does not
divide n, then n is 3k-perfect.
See Also: SigmaFunction, PerfectNumber
Related pages (outside of this work)