20170220, 17:21  #1 
Mar 2016
2·7·23 Posts 
amount of primes with p=n^2+1
A peaceful evening for all,
there is a comparison between 1) the amount of primes of all primes with p=n^2+1 and p  n^2+1 by their first appearance of the polynomial f(n)=n^2+1 (sieving from n=0 to n_max), 2) between the amount of those primes by their second appearance and 3) the amount of primes of p=n^2+1 by their first appearance. The last two amounts 2) and 3) seem to have nearly the same value. http://devalco.de/quadr_Sieb_x%5E2+1.php#4g By the way the 1) amount is infinite, which can be proved, the 2) amount is also infinite, the 3) amount seems also be infinite. This is not a complete mathematical proof, but a nice comparison between two amounts which have the same growing rate. For persons who are interested in prime sieving using the quadratic polynomial n^2+1 i recommand the link: http://devalco.de/quadr_Sieb_x%5E2+1.php Nice greetings from the primes Bernhard 
20170220, 18:08  #2 
Aug 2006
3^{2}·5·7·19 Posts 
At the moment it's not possible to prove that there are infinitely many primes of the form n^2 + 1, but it is possible to bound the number of n such that n^2 + 1 is prime using sieve theory.

20170221, 13:50  #3 
Feb 2017
Nowhere
10556_{8} Posts 
Regarding n^{2} + 1, n a positive integer, the closest result I know of is that the form represents infinitely many positive integers with at most two prime factors, or P_{2} integers:
Iwaniec, Henryk. Almostprimes represented by quadratic polynomials. Invent. Math. 47 (1978), no. 2, 171188. 
20170221, 15:00  #4 
Aug 2006
3^{2}·5·7·19 Posts 
I added a few results to the Wikipedia page on Landau's problems: the FriedlanderIwaniec theorem that there are infinitely many primes of the form x^2 + y^4 (where y^4 is a more permissive form of 1), Ankeny's conditional theorem that there are infinitely many primes of the form x^2 + y^2 with y = O(log x), and Deshouillers & Iwaniec's proof that gpf(x^2 + 1) > x^1.2 infinitely often.

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