prime, composite: An integer p > 1 is called a prime number, or simply a prime, if its only positive divisors are 1 and p; otherwise it is called composite.
Theorem 3.1: If p is prime and p | ab, then p|a or p|b.
Corollary 1: If p is prime and 
, then 
for some k, 
.
Corollary 2: If 
are all prime and 
, then 
for some k, .
Theorem 3.2 (Fundamental Theorem of Arithmetic): Every positive integer n>1 can be expressed as a product of primes uniquely (up to the order of the primes in the product).
Corollary: Any positive integer n>1 can be written uniquely in a canonical form
where, for i=1,2,...,r each 
is a positive integer and each 
is a
prime, with 
.
Theorem 3.3 (Pythagoras): 
is irrational.
Let's look at the alternative proof that our author suggests:
Pythagoras's theorem above is the one that purportedly caused one of his disciples his life: the hapless fellow disclosed the fact that there were these irrational numbers that couldn't be written as the ratio of integers, and other members of the school sent him to swim with the fishes... at least that's the story!;)
Hopefully you're well aware of these results: it's just now that we're seeing how they're deduced from simple principles, such as the well-ordering principle (there it is again!).