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Perfect numbers and Mersenne primes

It is remarkable that even at this ``elementary'' level there are many problems which are still unsolved. In this section, we mention one of the oldest unsolved problems in mathematics.

Let be an integer and let

For example, .
**Definition 1.6.13** *A ***perfect number** is an integer such that , in other words, is the sum of its proper divisors.
No odd perfect numbers are known. The following conjuecture may be the oldest unsolved problem in mathematics!

**Conjecture 1.6.14** *Odd perfect numbers don't exist.*

It is known if an odd perfect number exists then it must be at least .

**Lemma 1.6.15** *An integer is an even perfect number if and only if , where is a prime.*
Though this result is often quoted as being due to Euler, it may have been known to Euclid.

**proof**: We leave the ``if'' direction as an exercise.

``Only if'': Since is an even number, we can write , where is odd and . We know that , so let , with . Since , we have:

which gives us:
Therefore, is a divisor of and . But implies that is the sum of all of the divisors of which are less than (i.e. is the sum of a group of numbers which include ). This is possible only if the group consists of one number, and that number must be one. Therefore, , which implies that both and are prime. Letting gives the conclusion.
A prime number of the form is called a **Mersenne prime**. As we've seen already, it is unknown whether or not there are infinitely many Mersenne primes.

For further details on perfect numbers, see for example Ball and Coxeter [BC], page 66, or Hardy and Wright [HW], §16.8.

**Exercise 1.6.16** *Find all solutions to .*
**Exercise 1.6.17** *Show that there are no solutions to .*
**Exercise 1.6.18** *Show that if is multiplicative then so is .*
**Exercise 1.6.19** *Show that if , are multiplicative then so is .*
**Exercise 1.6.20** *Find all such that .*
**Exercise 1.6.21** *Show that is odd if and only if is a square.*
**Exercise 1.6.22** *Find all such that is odd.*
**Exercise 1.6.23** *Show that no perfect number is a square.*

**Exercise 1.6.24** *Show that if then .*

David Joyner 2007-09-03