There are an infinite number of primes! You knew that, but now you should be able to prove it.
A composite number a can be written as bc, where, WLOG, 
. If b
is prime, then, since 
, then a possesses a prime less than
; if not, then b contains a prime factor p, which must be less
than - and this factor must also be a prime factor of a, since
p|b, and b|a. It suffices then, to look for prime factors of a among the
primes 
.
Example: Determine whether 3731 is prime, or find its prime factorization.
The sieve of Eratosthenes is an interesting historical artifact: an early method for determining primes.
Example (homework): #2, p. 50.
Theorem 3.4 (Euclid): The primes are infinite in number.
Theorem 3.5: If 
is the 
prime, then 
.
Corollary: For 
, there are at least n+1 primes less than
.
Between 
and 2n there is at least one prime, from which one can show
that for ,
