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 2023-03-08, 09:23 #1 DF476   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
 2023-03-08, 10:02 #2 kruoli     "Oliver" Sep 2017 Porta Westfalica, DE 5×192 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…
 2023-03-08, 13:57 #3 Andrew Usher   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.
 2023-03-08, 20:46 #4 Dr Sardonicus     Feb 2017 Nowhere 22·32·181 Posts The question of whether there are infinitely many primes p = n2 + 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 Batemann-Horn 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 n2 + 1 has at most two prime factors: Henryk Iwaniec, Almost-Primes Represented by Quadratic Polynomials Inventiones mathematicae Volume: 47, page 171-188 (1978)
2023-03-08, 21:04   #5
VBCurtis

"Curtis"
Feb 2005
Riverside, CA

2×5×599 Posts

Quote:
 Originally Posted by Andrew Usher They have one already: generalised Fermat primes (excepting 2). The largest such primes known are therefore the largest GFNs.
Huh? Can you elaborate about how the largest known n^2 + 1 prime is therefore the largest GFN?

2023-03-08, 23:26   #6
science_man_88

"Forget I exist"
Jul 2009
Dartmouth NS

5·13·131 Posts

Quote:
 Originally Posted by VBCurtis Huh? Can you elaborate about how the largest known n^2 + 1 prime is therefore the largest GFN?
$(a^b)^c=a^{bc}=(a^c)^b$
$2^{2^n}+1=(2^{2^{n-1}})^2+1$

2023-03-08, 23:31   #7
charybdis

Apr 2020

22·3·5·19 Posts

Quote:
 Originally Posted by VBCurtis Huh? Can you elaborate about how the largest known n^2 + 1 prime is therefore the largest GFN?
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.

2023-03-08, 23:37   #8
science_man_88

"Forget I exist"
Jul 2009
Dartmouth NS

5×13×131 Posts

Quote:
 Originally Posted by charybdis 25*2^13719266+1
That is a Sierpinski number ( as written) not a Fermat

Last fiddled with by science_man_88 on 2023-03-08 at 23:41

2023-03-09, 00:42   #9
kar_bon

Mar 2006
Germany

22×13×59 Posts

Quote:
 Originally Posted by science_man_88 That is a Sierpinski number ( as written) not a Fermat
25*2^13719266+1 is a General Fermat Prime as stated in the Top5000, see also Proth primes for k=25 and this condition.

2023-03-09, 03:08   #10
charybdis

Apr 2020

22·3·5·19 Posts

Quote:
 Originally Posted by science_man_88 That is a Sierpinski number ( as written) not a Fermat
Firstly, as kar_bon pointed out, it is a generalized Fermat with exponent 2. Secondly, it's almost certainly not a Sierpinski number.

2023-03-09, 13:07   #11
Dr Sardonicus

Feb 2017
Nowhere

22×32×181 Posts

Quote:
Originally Posted by science_man_88
Quote:
 Originally Posted by charybdis 25*2^13719266+1
That is a Sierpinski number ( as written) not a Fermat
As written? That after-the-fact edit is the most inane attempt to whitewash a mistake I've seen in a long, long time - perhaps ever - on this Forum since I've been following it.

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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