# quadratic residues and prime divisor

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!

• These are 2 distinct questions, and you should post them separately. – Calvin Lin Mar 31 '14 at 2:54
• 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. – Ivan Loh Mar 31 '14 at 14:26

• 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. – Esteban Crespi Mar 31 '14 at 6:44