We finished our study of Book IX of Euclid's elements. To be precise we discussed the following two propositions:
This conclusion gives a way of computing the sum of the terms in the continued proportion as
If we denote the first term by a and the ratio of the terms by r, then this gives the familiar formula (you have seen it in Calculus 2 for sure, when you talked about the geometric series!)
What does this have to do with our discussion on prime numbers? That's where Proposition 36 comes into the picture. This final proposition is about (even) perfect numbers. It says that if the sum of the first p powers of 2
is a prime number, then the number M=s*2p-1 is perfect.
For instance if p=2 then s=1+2 = 3...thus M=2*3=6 is perfect
(that is the sum of its proper divisors: 6=1+2+3). Similarly,
if p=3, then s=1+2+22=7 is a prime so that M=7*4=28
is also perfect (indeed, 28=1+2+4+7+14). Please read carefully the commentaries:
the prime numbers of the form 2p-1 are called
Thus far only 43 of these numbers are known. Are the finite or infinite?
Euler showed that all even perfect numbers are of this form. What about the odd
perfect numbers? Do they exists?