Re: Number theory

On Sat, 29 Dec 2007, Chip Eastham wrote:
On Dec 29, 3:46 am, William Elliot <ma...@xxxxxxxxxxxxxxxxxx> wrote:
On Fri, 28 Dec 2007, Chip Eastham wrote:

The Goldbach conjecture remains unproven, though it is closely

for all even n > 2, some primes p,q with n = p + q

related to the twin prime conjecture, and sieving methods have
achieved partial success with both. Bertrand's postulate,
that there exists a prime p s.t. n < p < 2n for every integer
n > 1, is not hard to show once the Prime Number Theorem is
available, hence the connections to Tchebychev's and later
Erdos's "elementary" proofs of that result. However the
proposition that there exists a prime p s.t. n < p < n^2
for n > 1 is truly elementary and can be shown to high school
students by a bit of sieving techniques. Hence my doubts
that it suffices for the purpose here.

What's the prime number theorem?
The theorem by Erdos that was stated in Narco's post
or is it so complicated that I'd wish I hadn't asked?

for all n > 1, some prime p with n < p < n^2

Shucks, I'm not getting a handle on it.
Maybe if I sleep on it ...

AFAIK it remains unsolved whether there exists a prime p
s.t. n^2 < p < (n+1)^2 for every integer n > 1, i.e. a
prime between every pair of consecutive squares.

.