Largest Prime Factors of Polynomials

TL;DR

This paper proves that the largest prime factor of a specific quadratic product grows at least as fast as x^{3/2} for large x, improving previous bounds and enhancing understanding of prime factors in polynomial products.

Contribution

It establishes a new lower bound for the largest prime factor of a quadratic product, surpassing previous estimates and advancing number theory knowledge.

Findings

Largest prime factor p ≥ x^{3/2} as x→∞

Improves previous estimate p ≥ x^{1.279}

Enhances understanding of prime factors in polynomial products

Abstract

Let $x>1$ be a large number. This note shows that the largest prime factor of the quadratic product $\prod_{x\leq n\leq 2x}\left(n^2+1 \right)$ satisfies the relation $p \geq x^{3/2}$ as $x$ tends to infinity. This improves the current estimate $p \geq x^{1.279}$.