Prove that exist a many infinitely positive integers $n$ such that $n^2+1$ have a prime divisor greater than $2n + \sqrt{2n}$.

I was trying to solve but without interesting advances.

Any idea would be appreciated. Thanks in advance!

  • $\begingroup$ These are 2 distinct questions, and you should post them separately. $\endgroup$ – Calvin Lin Mar 31 '14 at 2:54
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    $\begingroup$ This is IMO 2008 Shortlist N6 (and also Question 3 on Day 1 of the contest); see page 50 of imo-official.org/problems/IMO2008SL.pdf for a solution. $\endgroup$ – Ivan Loh Mar 31 '14 at 14:26

We know that there are infinitely many numbers n such that n^2+1 is prime. And it is easy to prove that 2n+(2n)^0.5 < n^2+1.

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    $\begingroup$ As far as I know, we don't know if there are infinitely many integers $n$ such that $n^2+1$ prime, it is an open problem. $\endgroup$ – Esteban Crespi Mar 31 '14 at 6:44
  • $\begingroup$ It's known as Landau's fourth problem $\endgroup$ – punctured dusk Apr 15 '15 at 8:51

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