# Lattice Points in High-Dimensional Spheres

The Gauss Circle Problem is a classic open problem in number theory concerning the number of lattice points contained in a large circle.

Optimal error bounds are known in these approximations in a generalization of the Gauss Circle Problem to spheres in dimensions four and above.

In this post, I’ll give a purely analytic proof of this result for even dimensions greater than four, and explain why the method fails in the other cases.

# Prime-Generating Polynomials

Just how good are polynomials at producing primes? Do there exist polynomials that produce primes for arbitrarily many consecutive inputs?

In this post, I’ll give a brief overview of what we expect to be able to prove, and show how interpolating polynomials can produce record-breaking prime-generators. (And then break a record, because why not?)

# Calendars for the Modern Age

Our modern calendar, the Gregorian calendar, introduced several changes to the Julian calendar, including the introduction of a three-step algorithm for determining leap years.

But just how accurate is the Gregorian model? Can centuries of mathematics and astronomy lead us to a more accurate calendar?

# Sums of Squares and Density

Classification theorems of Euler, Lagrange, and Legendre describe the sets of integers that can be written as the sum of 2, 3, and 4 squares. In the last two cases, it follows easily that the density of these sets are 5/6 and 1.

The question of density is not so simple in the case of two squares. In this post, we resolve using an unexpected tool — Dirichlet’s theorem on primes in arithmetic progressions.

# A Generalization of Wilson’s Theorem (due to Gauss)

John Wilson (1741-1793) was a well-known English mathematician in his time, whose legacy lives on in his eponymous result, Wilson’s Theorem.

In this post, we’ll generalize Wison’s Theorem to non-prime modulus (and a few other generalizations) and give credit to Gauss for beating us to it by two hundred years.