 mersenneforum.org amount of primes with p=n^2+1
 Register FAQ Search Today's Posts Mark Forums Read 2017-02-20, 17:21 #1 bhelmes   Mar 2016 33·11 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   2017-02-20, 18:08 #2 CRGreathouse   Aug 2006 174A16 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.   2017-02-21, 13:50 #3 Dr Sardonicus   Feb 2017 Nowhere 22×1,049 Posts Regarding n2 + 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 P2 integers: Iwaniec, Henryk. Almost-primes represented by quadratic polynomials. Invent. Math. 47 (1978), no. 2, 171-188.   2017-02-21, 15:00 #4 CRGreathouse   Aug 2006 2·11·271 Posts I added a few results to the Wikipedia page on Landau's problems: the Friedlander-Iwaniec 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.  Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post wildrabbitt Hardware 3 2015-03-11 16:41 fivemack Hardware 5 2009-01-07 14:44 azhad Software 2 2004-10-16 16:41 dave_0273 Data 3 2003-11-01 17:07 David Software 1 2002-12-17 11:53

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