About k ^ 2 + 1 primes

[DF476]

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

[kruoli]

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…

[Andrew Usher]

They have one already: generalised Fermat primes (excepting 2). The largest such primes known are therefore the largest GFNs.

[Dr Sardonicus]

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)

[VBCurtis]

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?

[science_man_88]

Originally Posted by VBCurtis

Huh? Can you elaborate about how the largest known n^2 + 1 prime is therefore the largest GFN?

[charybdis]

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.

[science_man_88]

Originally Posted by charybdis

25*2^13719266+1

That is a Sierpinski number ( as written) not a Fermat

[kar_bon]

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.

[charybdis]

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.

[Dr Sardonicus]

Originally Posted by science_man_88

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.