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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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    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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