John Lundeberg

Mike Mitchaner

Mallory Moncivaiz

Summary:

This presentation will provide multiple classifications of classical numbers. Such numbers include perfect, amicable, and sociable numbers. It will provide many types of perfect numbers, the restrictions on even and odd perfect numbers, the correlation to Mersenne primes, and the conjectured infinitude of perfect numbers.

I. This section on *perfect numbers* includes definitions of perfect, almost perfect, superperfect, k perfect, multiplicative perfect, unitary perfect and hyperperfect numbers

a. Recall: *Perfect numbers* are positive integers n such that $\sigma(n)=2n$ where $\sigma(n)$ is the sum of the divisors function

b. *Almost Perfect Numbers* are positive integers such that $\sigma(n)=2n-1$

- The only known almost perfect numbers are powers of two
- The first few almost perfect numbers are 1,2,4,8, and 16
- It is still open to show that a number is almost perfect if and only if it is of the form 2
^{n} - Example: $\sigma(16)=1+2+4+8+16=32-1=31$

c. *Superperfect Numbers* are numbers for which $\sigma(\sigma(n))=2n$

- Even superperfect numbers are just 2
^{p-1}where M_{p}=2^{p}-1 is a Mersenne Prime - Example: $\sigma(\sigma(16))=\sigma(31)=31+1=32$

d. *K-Perfect Numbers* are positive integers such that $\sigma(n)=kn$

- Example: 120 is 3-Perfect! $\sigma(120)$=$\sigma$(2
^{3}3*5)=15*4*6=360=3*120

e. *Multiplicative Perfect Numbers* are numbers for which the product of divisors is equal to n^{2}

- Example: 6 is a multiplicative perfect number. Its divisors are 1,2,3,and 6 and their product is equal to 36=6
^{2}

f. *Unitary Perfect Numbers* are numbers n such that the sum of its proper unitary divisors not including itself is equal to n

- a is a proper unitary divisor of b if $(a,(b/a))=1$
- Example: 6 is a unitary perfect number. The divisors of 6 are 1,2,3,6 and we should check if 1,2, and 3 are proper unitary divisors

(1,6/1)=1, (2,6/2)=1, (3,6/3)=1 so they're all proper unitary divisors

so we have that the sum of proper unitary divisors of 6 =6

g. *Hyperperfect Numbers* are natural numbers n such that $n=1+k(\sigma(n)-n-1)$

- A number is perfect if it is 1-hyperperfect: $n=1+1(\sigma(n)-n-1)=\sigma(n)-n$
- Example: $6=1+1(\sigma(6)-6-1)$ where k=1

II. Amicable Numbers

- An amicable number are numbers m and n such that $\sigma(m)=\sigma(n)=m+n$
- In every known case the numbers in an amicable pair are either both even or both odd, it is open to show that this is always the case
- Example: 1184 and 1210 form an amicable pair

$\sigma(1184)$=$\sigma$(2^{5}3*7)