# Counting Function for Polygonal Numbers of Rank Greater Than 2.

1. Feb 7, 2010

### Don Blazys

Just as $$Li(x)$$ can be used to approximate how many
prime numbers there are under a given number $$x$$,
my function $$B(x)$$, which you can find here:

http://donblazys.com/on_polygonal_numbers.pdf

can be used to approximate how many
polygonal numbers of rank greater than $$2$$
there are under a given number $$x$$.

What makes this function really interesting is that
it seems to require a constant $$\alpha$$
that either is, or is very, very close to
the actual "experimental value" of the fine structure constant !.

Thus, this function presents us with a wonderful mystery.

Is the actual fine structure constant required
in order to achieve the best possible approximation
(meaning the lowest possible upper bound)
for how many polygonal numbers of rank greater than $$2$$
there are under a given number $$x$$ ?

Don.

Last edited: Feb 7, 2010