2025/02/02 by Thomas Ehrenborg, Ehrenborg, Thomas
Mathematics · #FOS: Mathematics #General Mathematics (math.GM) #History and Theory of Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2502.06804
openalex publication_date 2025/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 πr2, 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 ≤ n. The Prime Number Theorem predicts that the number of primes less than or equal to n is asymptotic to (n)/(log n). We find that the number of Gauss Circle Primes for r ≤ n is also of order (n)/(log n) for n ≤ 2 × 106. We include a heuristic argument that the Gauss Circle Primes can be approximated by (n)/(log n).