Quadratic polynomials at prime arguments

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.

17 pages, 1 figure. Minor changes, in Mathematische Zeitschrift, 2016

References in corpus (1)

• Almost-prime values of polynomials at prime arguments

Cited by in corpus (2)

• On the Least Common Multiple of Polynomial Sequences at Prime Arguments
• Almost-prime values of reducible polynomials at prime arguments