Intro Chapter

So I am pretty excited.  I had originally bought the book How to Prove It though Google online books. Unfortunately it was missing the intro chapter and there were a bunch other issues that made using it a pain (thought the fact that I have it on my iPad is cool since I pretty much have my iPad with me at all times).

Anyway I just finished reading the Intro chapter and have started working on some of the problems and I have to say, this book is so freaking cool.  Just the exposure to some really cool math stuff that I have never seen before is totally worth the cover cost.

Couple highlights.  So there are prime numbers of the form $$ 2^n-1 $$ where n is a prime.  They are called Mersenne primes after a guy called, well of course by a monk (I think he was a monk now I am not so certain) named Mersenne (1588-1647).  Mersenne primes are currently some of the largest prime number numbers in fact, the largest know prime is a Mersenne prime.

Anyway, that's kind of cool in a did you know this minutia kind of way.  Here is the cool thing our buddy Euclid proved that if a prime number is of the form $$ 2^n-1 $$ then $$ 2^{n-1}(2^n-1) $$ is perfect.  Meaning every Mersenne prime can be used to generate a perfect number.

Note a perfect number is a number n such that the sum of all its factors less than n = n.  So for example 28 is a perfect number because its factors: 1,2,4,7,14 summed up equal 28.

What really got me though was later on Euler (1707-1783) proved that every perfect number is generated this way.  So again using 28 as an example.  $$28 = 2^2(2^3-1) $$.

So those were all pretty cool things.  Here is something cool you can do with non prime numbers of the form $$ 2^n-1 $$ .  If n is not prime then as a consequence $$ 2^n-1 $$ is also not prime.  As a result we can take non Mersenne prime numbers of that form and factor them (if of course you know how to write them as $$ 2^n-1 $$ which I will agree is no mean feet and something I would really like to know how to do - maybe this would be a nice little programming project in the very near future.

Anyway, lets take a perfect number of this form, for example the one given in the book: 32,767 which equals $$ 2^{15}-1 $$, then we can two of its factors.

$$ 2^{15}-1 = 2^{3 \cdot 5}-1 $$
$$ = (2^3-1)(1+2^{3}+2^{3\cdot2}+...+2^{3\cdot4} $$

The general formula for this is:  if $$n=a\cdot b $$ then $$2^n-1= 2^{a \cdot b}-1$$
$$ = (2^a-1)(1+2^a+2^{a \cdot 2}+...+2^{a \cdot (b-1)}) $$

Pretty cool right.