Small Pythagorean triples modulo prime powers

TL;DR

This paper derives an asymptotic formula for small solutions to quadratic congruences involving Pythagorean triples modulo prime powers, focusing on solutions with bounded size relative to the prime power.

Contribution

It provides a new asymptotic estimate for small Pythagorean triples modulo prime powers, extending understanding of quadratic congruences in this setting.

Findings

Asymptotic formula for solutions with size up to p^{ u n}

Results valid for primes p > 5 and u > 1/2

Enhanced understanding of quadratic congruences modulo prime powers

Abstract

Let $p>5$ be a fixed prime. We obtain an asymptotic formula related to small solutions of quadratic congruences of the form $x_1^2+x_2^2\equiv x_3^2\bmod{p^n}$ where $\max\{|x_1|,|x_2|,|x_3|\}\le p^{\nu n}$ with $\nu>1/2$.