On the greatest prime factor and uniform equidistribution of quadratic polynomials

TL;DR

This paper proves that the greatest prime factor of quadratic polynomials grows rapidly infinitely often and establishes uniform distribution results for roots of quadratic congruences, advancing understanding in prime factorization and distribution.

Contribution

It provides an unconditional lower bound on the greatest prime factor of quadratic polynomials and demonstrates uniformity in distribution under natural hypotheses.

Findings

Greatest prime factor of n^2+h is at least n^{1.312} infinitely often

Uniformity in h q n^{1+o(1)} under a natural hypothesis

Uniform distribution of roots of quadratic congruences modulo primes

Abstract

We show that the greatest prime factor of $n^2+h$ is at least $n^{1.312}$ infinitely often. This gives an unconditional proof for the range previously known under the Selberg eigenvalue conjecture. Furthermore, we get uniformity in $h \leq n^{1+o(1)}$ under a natural hypothesis on real characters. The same uniformity is obtained for the equidistribution of the roots of quadratic congruences modulo primes. We also prove a variant of the divisor problem for $ax^2+by^3$, which was used by the second author to give a conditional result about primes of that shape.