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