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Primes of the form $n^2+1$ - hard?

$1, 2, 5, 10, 17, \ldots$

Are there an infinite number of primes in this sequence $1 + t^2$, $t$ being an integer?

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    This is the 4th one among Landau's problems. All of them are currently open. – Srivatsan Sep 14 '11 at 13:20
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    Of course the answer is yes - it's just that no one has a good idea about how to prove it. – Gerry Myerson Sep 14 '11 at 13:24
  • Re @Gerry's comment, but slightly off the tangent. Is there any polynomial $p$ of degree $> 1$ for which we can prove infinitude of primes of the form $p(t)$? In other words, is there any intrinsic difficulty with this particular choice $t^2+1$? I feel the answer is no, but confirmation will be nice. – Srivatsan Sep 14 '11 at 13:31
  • Copied answer from duplicate thread: This is an incredibly difficult problem.

    It is one of Landau's 4 problems which were presented at the 1912 international congress of mathematicians, all of which remains unsolved today nearly 100 years later.

     – Eric Naslund Sep 14 '11 at 13:40
  • @Srivatsan, no, there is no one-variable polynomial of degree exceeding 1 for which it has been proved that the polynomial represents infinitely many primes. My guess, which isn't really worth very much, is that if the problem is ever settled it will be settled for all polynomials; there won't be anything special about $x^2+1$. – Gerry Myerson Sep 14 '11 at 13:45
  • Thanks for the clarification, @Gerry. To those who are interested (in this off-tangent discussion ;)), Bunyakovsky's conjecture is a generalization of this problem for arbitrary polynomials. – Srivatsan Sep 14 '11 at 13:52
  • Do we at least know the answer to the following question: does there exist a positive integer $k$ such that $n^2 + k$ represents infinitely many primes (here $n$ varies over the positive integers)? – Mark Sep 14 '11 at 14:51

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