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I know it is unknown if there are infinitely many primes of the form $n^2+1$. Is it known if there is a positive integer $k$ such that $|\{n\in\mathbb{Z}:n^2+1 \text{ has at most k prime factors}\}|=\infty$?

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up vote 14 down vote accepted

Yes, Iwaniec, "Almost-primes represented by quadratic polynomials", Inventiones Math., 47:171–188, 1978, proves that there exist infinitely many integers $n$ such that $n^2 + 1$ is either prime or the product of two primes.

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Andres defeated me by 20seconds – anonymous Nov 19 '10 at 23:07
I'm teaching number theory, so I had the reference in my desk. :-) – Andrés E. Caicedo Nov 19 '10 at 23:11
Oh, i googled it and got a reference. Nice to know anyway. – anonymous Nov 19 '10 at 23:14

Please see Iwaniec, "Almost-primes represented by quadratic polynomials", Inventiones Math

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