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primes of form n^2 + 1

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  • WarrenS
    A famous old conjecture is that infinitely many primes of form n^2+1 exist. I have no idea how to settle that, but can one instead settle this easier problem:
    Message 1 of 1 , Oct 26, 2012
      A famous old conjecture is that infinitely many primes of form n^2+1
      exist.

      I have no idea how to settle that, but can one instead settle this easier problem:
      are there infinitely many n such that n^2+1 is either prime or "P2"
      (product of two primes)?

      The answer is YES!

      From
      Harald A. Helfgott: On the square-free sieve, Acta Arithmetica 115,4 (2004) 349-402
      http://arxiv.org/abs/math/0309109
      also citing
      T. Estermann: Einige S"atze "uber quadratfreie Zahlen, Math. Annalen 105 (1931) 653-662.
      we know that there are infinitely many n^2+1 which are squarefree.

      From
      Henryk Iwaniec: Almost-primes represented by quadratic polynomials
      Inventiones math. 47,2 (1978) 171-188
      http://www.springerlink.com/content/q577674g12011317/fulltext.pdf
      we know that any polynomial P(n) that is not trivially excluded
      represents an infinity of numbers that are products of <=k primes,
      where k=degP + const*log(degP).

      Finally, Iwaniec as his Main Theorem shows there are an infinite set of n such
      that n^2+1 is a product of at most two primes. Also works for 4*n^2+1 and
      indeed for any irreducible quadratic P(n) with P(0)=1, and the count of such n
      below X is show to grow eventually at last proportionally to X/logX;
      proportionality constant depends on P in a simple way.
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