matrimc-ji, Amber G.ji, Thanks for your kind words. I did not know that this link was given earlier; sorry about the repetition.

Now, something a little more substantive(analytical):

There are more primes than there are perfect squares. In fact, in a meaning that can be expressed more precisely, there are

many, many, many more primes than there are perfect squares.

The way to make this precise is through the following two lemmas:

Lemma 1: The harmonic series diverges.

That is to say, the series 1 + 1/2 + 1/3 + 1/4 + 1/5 + ....... diverges.

Proof:

Easiest way to see is to visualize this sum as a triangles of heights 1, 1/2, 1/3, .... placed on the x-axis, next to next, of width 1, and this can be bound from below by the integral of (1/x )dx from 1 to infinity.

This proof with illustration and more details, and also another proof using the comparison test, are available at:

http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#Divergence.

Lemma 2: The series 1 + 1/4 + 1/9 + 1/16 + 1/25 + ..... , i.e., the sum of reciprocals of the perfect squares,

converge.

Proof:

http://en.wikipedia.org/wiki/Basel_problem#The_Riemann_zeta_functionSo, now we have enough background to make our theorem interesting:

Theorem: The sums of reciprocals of primes diverges.

Proof:

http://en.wikipedia.org/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes#Proofs ___________________________________________________

It is in this sense that there are many, many, many more primes than there are perfect squares. Please note that this whole line of thought is related to the

Riemann Zeta function which in fact came from a "proof" of Euler that there are infinitely many primes. Maybe let it be for another day. Btw, all this math typing in plain ASCII is bit restricting. Is there a possibility for using something like jsMath or MathJax in the forum(a long shot; still just noting the possibility).