The bonus material from yesterday:

Perfect numbers are those numbers which are equal to the sum of their divisors (other than themselves). For instance, 6 is divisible by 1, 2 and 3 (and 6, but we don’t include this in what follows). If you add these together, you get 6. 6 is a perfect number. 8 is divisible by 1, 2 and 4. Add these together and you get 7 – so 8 is not a perfect number (actually it is called deficient, the other alternative is an abundant number).

It was shown by Euclid that $2^{p-1}(2^p-1)$ is an even perfect number whenever $2^p-1$ is a prime (actually a prime of this form is called a Mersenne prime). In fact, it was shown by Euler two millennia later that every even perfect number could be written in this form! This is known as the Euclid-Euler theorem. So far there are 48 Mersenne primes known, and therefore 48 even perfect numbers known. Might we have found the largest? We just don’t know.

In fact in the formula for a Mersenne prime $2^p-1$, $p$ itself is a prime number. The largest Mersenne prime so far found is: $2^{57885161}-1$

We still don’t know if there are an infinite number of Mersenne primes or, equivalently if there are an infinite number of even perfect numbers.

We also don’t know whether there are ANY perfect odd numbers.

Both the Mersenne problem and the odd perfect number problem are unsolved questions in number theory.

Infinities

I mentioned today in class that there were some weird properties of infinities…in fact infinity is weirder than you could possibly imagine. Take a look here for a few of the weird paradoxes that show up…

 How clear is this post?