20230308, 09:23  #1 
Mar 2023
1 Posts 
About k ^ 2 + 1 primes
I have created a little program to find the prime numbers of k ^ 2 + 1, where k is an even number (except 1). And I have been surprised by how little information there is about them. I have only found one entry in the OEIS that barely reaches 40 numbers. So I ask, is there a search project for these primes? Or do they have a name (I have thought of calling them Asfolk)? Do they have a pattern?
Thanks for reading me 
20230308, 10:02  #2 
"Oliver"
Sep 2017
Porta Westfalica, DE
5×19^{2} Posts 
It is unlikely there is a search for them  they do not grow remotely fast enough. They have been looked at as early as by Leonhard Euler in De numeris primis valde magnis (as stated in OEIS A002496). A table with 10,000 elements is available here.
Even if you start with "big" numbers and look for primes, they have no form which would make them easy to prove prime, also they are not easier to sieve than others AFAICT. Of course you could define a name for them… 
20230308, 13:57  #3 
Dec 2022
2×3×7×13 Posts 
They have one already: generalised Fermat primes (excepting 2). The largest such primes known are therefore the largest GFNs.

20230308, 20:46  #4 
Feb 2017
Nowhere
2^{2}·3^{2}·181 Posts 
The question of whether there are infinitely many primes p = n^{2} + 1 is one of four problems about primes listed by Edmund Landau at the 1912 International Congress of Mathematicians. All four are still unsolved.
The BatemannHorn Conjecture for prime evaluations of polynomials gives an asymptotic formula which is consistent with the limited numerical evidence. It is known that there are infinitely many n for which n^{2} + 1 has at most two prime factors: Henryk Iwaniec, AlmostPrimes Represented by Quadratic Polynomials Inventiones mathematicae Volume: 47, page 171188 (1978) 
20230308, 21:04  #5 
"Curtis"
Feb 2005
Riverside, CA
2×5×599 Posts 

20230308, 23:26  #6 
"Forget I exist"
Jul 2009
Dartmouth NS
5·13·131 Posts 

20230308, 23:31  #7 
Apr 2020
2^{2}·3·5·19 Posts 
Numbers of the form n^2+1 are GFNs since 2 is a power of 2. The Top 5000 lists them as such, e.g. 25*2^13719266+1. In fact GFNs and numbers of the form n^2+1 are one and the same.

20230308, 23:37  #8 
"Forget I exist"
Jul 2009
Dartmouth NS
5×13×131 Posts 
That is a Sierpinski number ( as written) not a Fermat
Last fiddled with by science_man_88 on 20230308 at 23:41 
20230309, 00:42  #9 
Mar 2006
Germany
2^{2}×13×59 Posts 
25*2^13719266+1 is a General Fermat Prime as stated in the Top5000, see also Proth primes for k=25 and this condition.

20230309, 03:08  #10 
Apr 2020
2^{2}·3·5·19 Posts 
Firstly, as kar_bon pointed out, it is a generalized Fermat with exponent 2. Secondly, it's almost certainly not a Sierpinski number.

20230309, 13:07  #11  
Feb 2017
Nowhere
2^{2}×3^{2}×181 Posts 
Quote:
In the first place, how a number is written doesn't affect whether it's a Sierpinski number, or a GFN. The number 25*2^13719266 + 1 clearly is (5*2^6859633)^2 + 1, thus a GFN. In the second place, k is a Sierpinski number when k*2^n + 1 is composite for every positive integer n. How do you claim to know that k = 25*2^13719266 + 1 has this property? I'm guessing you don't know any such thing. In trying to whitewash one mistake, you've made another. 

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