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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](http://en.wikipedia.org/wiki/Landau's_problems). All of them are currently open.2011-09-14
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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.2011-09-14
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    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.2011-09-14
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    Copied answer from duplicate thread: This is an *incredibly* difficult problem. It is one of [Landau's 4 problems][1] which were presented at the 1912 international congress of mathematicians, all of which remains unsolved today nearly 100 years later. [1]: http://en.wikipedia.org/wiki/Landau%27s_problems2011-09-14
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    @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 $x^2+1$.2011-09-14
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    Thanks for the clarification, @Gerry. To those who are interested (in this off-tangent discussion ;)), [Bunyakovsky's conjecture](http://en.wikipedia.org/wiki/Bunyakovsky_conjecture) is a generalization of this problem for arbitrary polynomials.2011-09-14
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    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)?2011-09-14

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