Go Back > Math Stuff > Computer Science & Computational Number Theory

Thread Tools
Old 2017-02-20, 17:21   #1
bhelmes's Avatar
Mar 2016

33·11 Posts
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.

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:

Nice greetings from the primes
bhelmes is online now   Reply With Quote
Old 2017-02-20, 18:08   #2
CRGreathouse's Avatar
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.
CRGreathouse is offline   Reply With Quote
Old 2017-02-21, 13:50   #3
Dr Sardonicus
Dr Sardonicus's Avatar
Feb 2017

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.
Dr Sardonicus is offline   Reply With Quote
Old 2017-02-21, 15:00   #4
CRGreathouse's Avatar
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.
CRGreathouse is offline   Reply With Quote

Thread Tools

Similar Threads
Thread Thread Starter Forum Replies Last Post
amount of memory when P-1 factoring wildrabbitt Hardware 3 2015-03-11 16:41
A certain amount of mild vexation fivemack Hardware 5 2009-01-07 14:44
Choosing amount of memory azhad Software 2 2004-10-16 16:41
Optimum amount of RAM for P-1 testing dave_0273 Data 3 2003-11-01 17:07
Amount of factoring David Software 1 2002-12-17 11:53

All times are UTC. The time now is 02:16.

Thu Jan 28 02:16:37 UTC 2021 up 55 days, 22:27, 0 users, load averages: 2.80, 2.60, 2.61

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2021, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.