Equidistribution for Solutions of $p+m^2+n^2=N$, and for Ch\^{a}telet Surfaces

TL;DR

This paper introduces a new method to analyze the equidistribution of solutions to equations involving sums of squares and applies it to the specific case of primes plus two squares equaling a fixed number.

Contribution

It develops a general approach for equidistribution problems involving quadratic forms and demonstrates its effectiveness on the equation p + m^2 + n^2 = N.

Findings

Established a new method for equidistribution analysis.

Applied the method to primes plus squares problem.

Provided insights into solutions distribution for Ch ext{"a}telet surfaces.

Abstract

We present a general method for handling problems that ask for the equidistribution of solutions to equations involving $m^2+n^2$, and illustrate it by considering $p+m^2+n^2=N$.