**How many perfect numbers are there ?**

Looks like Aubrey liked the entry about the prime numbers :-) so let me now talk about the so called perfect numbers. As usual we have to know what we are talking about...

For instance 4 is not perfect. The divisors of 4 (other than 4) are 1 and 2, and 1+2 = 3. Another instance of non perfect number is 12. We have that the divisors of 12 are 1, 2, 3, 4 and 6, and we have 1+2+3+4+6 = 16. Note how the sum of the divisors of 4 is a number smaller than 4 and the sum of the divisors of 12 is a number greater than 12. In French we have words for those two cases, but I don't remember them right now (and don't want to look it up).

6 is a perfect number, because the divisors of 6 are 1,2 and 3 and 6 = 1+2+3. The three other perfect numbers (that I know by heart) are 28, 496, and 8128. We (mathematicans) know few more of them (which my computer is still trying to find out -- I think that my program will crash out of memory before finding the fifth one...).

Coming back to the question, we don't know whether there are an infinite number of them. And we have never found any odd perfect number (all the ones I know and have heard of are even).

**Definition**: an integer strictly greater than 1 is said*perfect*if it is the sum of all its divisors (in which we include 1, but not the number itself).For instance 4 is not perfect. The divisors of 4 (other than 4) are 1 and 2, and 1+2 = 3. Another instance of non perfect number is 12. We have that the divisors of 12 are 1, 2, 3, 4 and 6, and we have 1+2+3+4+6 = 16. Note how the sum of the divisors of 4 is a number smaller than 4 and the sum of the divisors of 12 is a number greater than 12. In French we have words for those two cases, but I don't remember them right now (and don't want to look it up).

6 is a perfect number, because the divisors of 6 are 1,2 and 3 and 6 = 1+2+3. The three other perfect numbers (that I know by heart) are 28, 496, and 8128. We (mathematicans) know few more of them (which my computer is still trying to find out -- I think that my program will crash out of memory before finding the fifth one...).

Coming back to the question, we don't know whether there are an infinite number of them. And we have never found any odd perfect number (all the ones I know and have heard of are even).