Almost-Primes Represented by Quadratic Polynomials

TL;DR

This paper extends Iwaniec's result by proving that any irreducible quadratic polynomial meeting certain conditions takes on infinitely many values with at most two prime factors, generalizing the earlier specific case of n^2 + 1.

Contribution

It generalizes Iwaniec's theorem to a broader class of irreducible quadratic polynomials under certain hypotheses.

Findings

Quadratic polynomials with certain properties take infinitely many almost-prime values.

The proof follows Iwaniec's approach for the specific polynomial n^2 + 1.

The result applies to a wider class of irreducible quadratic polynomials.

Abstract

In his paper Almost-Primes Represented by Quadratic Polynomials, Iwaniec proved that the polynomial n^2 + 1 takes on values with at most two prime factors (counted with multiplicity) infinitely often. He states that "in order to avoid technical complications, we shall restrict our proof to the polynomial n^2 + 1.". In this exposition, we follow Iwaniec's proof and show that for any irreducible quadratic polynomial G(n) (satisfying some obviously necessary hypotheses), G(n) has at most two prime factors for infinitely many values of n.