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Old 2017-02-20, 17:21   #1
bhelmes
 
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Mar 2016

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Default 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
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Old 2017-02-20, 18:08   #2
CRGreathouse
 
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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.
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Old 2017-02-21, 13:50   #3
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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.
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Old 2017-02-21, 15:00   #4
CRGreathouse
 
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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.
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