Lecture 20


We finished our study of Book IX of Euclid's elements. To be precise we discussed the following two propositions: The first proposition sounds a little misterious, but (in plain English) it says that if a sequence of numbers a1, a2, a3, ..., an, an+1 is in continued proportion, i.e.,

a1:a2 = a2:a3 = ... = an:an+1

then

(a2 – a1) : a1 = (an+1 – a1) : (a1 + a2 + ... + an).

This conclusion gives a way of computing the sum of the terms in the continued proportion as

a1 + a2 + ... + an = a1 an+1 – a1
a2 – a1 		.

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!)

a + ar + ar2 + ... + arn-1 = a rn – 1
r – 1 		.

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

s = 1 + 2 + 22 + ... + 2p-1 (= 2p-1)

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 Mersenne primes. 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?

Two other famous conjectures we discussed in class related to prime numbers are: Goldback Conjecture and the Twin Primes Conjecture.

Click on the above links to go directly to the proofs and commentaries of the results.