On the Discrepancy of the Roots of $x^2+1$ and $x^2+2$ to Prime Moduli

TL;DR

This paper explores the distribution of roots of quadratic polynomials modulo primes, proposing a new conjecture and analyzing their behavior under the assumption of the Bateman-Horn conjecture.

Contribution

It introduces a new conjecture related to the roots of specific quadratic polynomials modulo primes and studies their distribution assuming major conjectures.

Findings

Proposes a conjecture linking roots of quadratics to prime distributions

Analyzes roots of x^2+1 and x^2+2 under conjectural assumptions

Utilizes Erdős-Turán-Koksma inequality in the analysis

Abstract

In this paper, we make a conjecture (conjecture 1) related to the Bateman-Horn conjecture and proceed to study the roots of $x^2+1$ and $x^2+2$ to prime moduli, assuming the truth of the Bateman-Horn conjecture and conjecture 1 and using the Erd\H{o}s-Turan-Koksma inequality.