Quadratic polynomials at prime arguments

TL;DR

This paper proves that for a fixed quadratic irreducible polynomial with certain properties, infinitely many prime arguments produce values with at most four prime factors, and the greatest prime factor of these values exceeds p^{0.847} infinitely often.

Contribution

It improves previous results by reducing the prime factors count from 5 to 4 and establishes a new lower bound on the greatest prime factor of polynomial values at primes.

Findings

Infinitely many primes p with f(p) having at most 4 prime factors.

Infinitely often, P^+(f(p)) > p^{0.847}.

Enhanced understanding of prime values of quadratic polynomials.

Abstract

For a fixed quadratic irreducible polynomial $f$ with no fixed prime factors at prime arguments, we prove that there exist infinitely many primes $p$ such that $f(p)$ has at most 4 prime factors, improving a classical result of Richert who requires 5 in place of 4. Denoting by $P^+(n)$ the greatest prime factor of $n$, it is also proved that $P^+(f(p))>p^{0.847}$ infinitely often.