Gauss Circle Primes

TL;DR

This work introduces Gauss Circle Primes, primes among the lattice-point counts $C(r)$ in Gauss's circle problem, and studies their distribution up to $n$. It combines classical circle-counting bounds with a probabilistic model to argue that the Gauss Circle Prime count $\kappa(n)$ grows on the order of $\frac{n}{\log n}$, corroborated by data showing $\pi(n) > \kappa(n) > \frac{n}{\log n}$ for large $n$ and $\kappa(n)$ closely tracking $\pi(n)$. A heuristic derivation links $\kappa(n)$ to the prime-counting function by considering $C(k) \approx \pi k^2$ and the oddness of $C(k)$, explaining why $\kappa(n)$ should mirror $\pi(n)$ asymptotically. The paper also outlines future directions, including twin Gauss Circle Primes, infinitude conjectures, connections to Skewes-type behavior, and higher-dimensional analogues, highlighting potential implications for number theory and cryptography.

Abstract

Given a circle of radius $r$ centered at the origin, the Gauss Circle Problem concerns counting the number of lattice points $C(r)$ within this circle. It is known that as $r$ grows large, the number of lattice points approaches $πr^2$, that is, the area of the circle. The present research is to study how often $C(r)$ will return a prime number of lattice points for $r \leq n$. The Prime Number Theorem predicts that the number of primes less than or equal to $n$ is asymptotic to $\frac{n}{\log n}$. We find that the number of Gauss Circle Primes for $r \leq n$ is also of order $\frac{n}{\log n}$ for $n \leq 2 \times 10^6$. We include a heuristic argument that the Gauss Circle Primes can be approximated by $\frac{n}{\log n}$.

Figures (2)

• Figure 1: Lattice points on a circle of radius $5$ showing $C(5) = 81$.
• Figure 2: Geometric argument for Gauss' error bound.

Theorems & Definitions (7)

• Theorem 1.1: Gauss 1834
• proof
• Theorem 1.2
• Theorem 1.3: Hadamard and de la Vallée Poussin, 1896
• Theorem 2.1
• proof
• proof