In my last post, I mentioned I would post my article proper on WordPress. Someone then told me about latex2wp, a python script that will translate a tex file into something postable on WordPress. So I did it, and it works pretty well! Other than changing references (removing them) and a few stylistic things here and there, and any \begin{align} type environments, it works perfectly.

So here it is:

In this, I present a method of quickly counting the number of lattice points below a quadratic of the form . In particular, I show that knowing the number of lattice points in the interval , then we have a closed form for the number of lattice points in any interval . This method was inspired by the collaborative Polymath4 Finding Primes Project, and in particular the guidance of Dr. Croot from Georgia Tech.

**1. Intro **

Suppose we have the quadratic . In short, we seperate the lattice points into regions and find a relationship between the number of lattice points in one region with the number of lattice points in other regions. Unfortunately, the width of each region is , so that this does not always guarantee much time-savings.

This came up while considering

In particular, suppose we write , so that we have . Then, expanding like , we see that

And correspondingly, we have that

Now, I make a great, largely unfounded leap. This is *almost* like a quadratic, so what if it were? And then, what if that quadratic were tremendously simple, with no constant nor linear term, and with the only remaining term having a rational coefficient? Then what could we do?

**2. The Method **

We want to find the number of lattice points under the quadratic in some interval. First, note that

Then we can sum over an interval of length q, and we’ll get a relationship with the next interval of length q. In particular, this means that

Now I adopt the notation , so that we can rewrite equation 5 as

Of course, we quickly see that we can write the right sum in closed form. So we get

We can extend this by noting that , so that

Extending to multiple intervals at once, we get

So, in short, if we know the number of lattice points under the parabola on the interval , then we know in time the number of lattice points under the parabola on an interval .

Unfortunately, when I have tried to take this method back to the Polymath4-type problem, I haven’t yet been able to reign in the error terms. But I suspect that there is more to be done using this method.