Largest Prime Factors of Polynomials

Abstract

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

Eight Pages. Keywords: Prime number; Distribution of prime; Polynomial prime value