## Prime sieving on the polynomial f(n)=n^2+1

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• A beautiful day, There are some results for primes concerning the polynom f(n)=n^2+1 109.90.3.58/devalco/quadr_Sieb_x^2+1.htm Besides some nice algorithms how
Message 1 of 4 , Jul 4, 2012
A beautiful day,

There are some results for primes concerning the polynom f(n)=n^2+1

Besides some nice algorithms how to calculate these primes,
there are results for n up to 2^40

You will find some nice graphics and some calculations.

Mathematical feedback is welcome :-)

Greetings from the primes
Bernhard
• Hello Bernhard, Looking at table 4a, it seems to present column c as the number of primes up to 2^n, and column d as the number of primes of the form x^2+1 up
Message 2 of 4 , Jul 5, 2012
Hello Bernhard,

Looking at table 4a, it seems to present column c as the number of primes up to 2^n, and column d as the number of primes of the form x^2+1 up to 2^n. But the numbers can't be correct so am I misunderstanding something?

I'm looking at the small table 5b where it seems you extrapolate the number of primes of the form x^2+1 which are between 2^n and 2^(n+1). The number of such primes is decreasing as n increases, but it seems to me it should be going up.

You wrote in the conclusion,

"The density of the distribution of the primes p=x^2+1 goes surely to zero.
Supposing that the exploration of the second scenario is rigth, it could be supposed that the primes of the form x^2+1 are limited."

Are you saying that the number of primes of the form x^2+1 appears to be finite?

Thanks,

Mark

--- In primenumbers@yahoogroups.com, "bhelmes_1" <bhelmes@...> wrote:
>
> A beautiful day,
>
> There are some results for primes concerning the polynom f(n)=n^2+1
>
>
> Besides some nice algorithms how to calculate these primes,
> there are results for n up to 2^40
>
> You will find some nice graphics and some calculations.
>
> Mathematical feedback is welcome :-)
>
> Greetings from the primes
> Bernhard
>
• The message below got stuck in the ether for over a day, and surely the Higgs boson had something to do with it. Anyways, Bernhard has since clarified things
Message 3 of 4 , Jul 7, 2012
The message below got stuck in the ether for over a day, and surely the Higgs boson had something to do with it. Anyways, Bernhard has since clarified things for me privately and I should be better now.

Mark

--- In primenumbers@yahoogroups.com, "Mark" <mark.underwood@...> wrote:
>
>
> Hello Bernhard,
>
> Looking at table 4a, it seems to present column c as the number of primes up to 2^n, and column d as the number of primes of the form x^2+1 up to 2^n. But the numbers can't be correct so am I misunderstanding something?
>
> I'm looking at the small table 5b where it seems you extrapolate the number of primes of the form x^2+1 which are between 2^n and 2^(n+1). The number of such primes is decreasing as n increases, but it seems to me it should be going up.
>
>
> You wrote in the conclusion,
>
> "The density of the distribution of the primes p=x^2+1 goes surely to zero.
> Supposing that the exploration of the second scenario is rigth, it could be supposed that the primes of the form x^2+1 are limited."
>
> Are you saying that the number of primes of the form x^2+1 appears to be finite?
>
> Thanks,
>
> Mark
>
>
>
> --- In primenumbers@yahoogroups.com, "bhelmes_1" <bhelmes@> wrote:
> >
> > A beautiful day,
> >
> > There are some results for primes concerning the polynom f(n)=n^2+1
> >
> >
> > Besides some nice algorithms how to calculate these primes,
> > there are results for n up to 2^40
> >
> > You will find some nice graphics and some calculations.
> >
> > Mathematical feedback is welcome :-)
> >
> > Greetings from the primes
> > Bernhard
> >
>
• Hello Mark ... i distinguish between primes p=x^2+1 and the primes which appear as divisor for the first time p | x^2+1 and p
Message 4 of 4 , Jul 7, 2012
Hello Mark

> Looking at table 4a, it seems to present column c as the number of primes up to 2^n, and column d as the number of primes of the form x^2+1 up to 2^n. But the numbers can't be correct so am I misunderstanding something?

i distinguish between primes p=x^2+1 and
the primes which appear as divisor for the first time
p | x^2+1 and p < x^2+1

for x<=2^5 = 32 i get
p=x^2+1 : 2(x=1), 5(x=2), 17(x=4), 37(x=6), 101(x=10), 197(x=14), 257(x=16), 401(x=20), 577(x=24), 677(x=26),
number of Primes for x<32 is 10 as table 4a) column D indicates

p|x^2+1 : 13(x=5), 41(x=9), 61(x=11), 29(x=12), 113(x=15), 181(x=19),
97(x=22), 53(x=23), 313 (x=25), 73(x=27), 157(x=28), 421(x=29)
number of Primes for x<=32 is 12 as table 4a) column E indicates

> I'm looking at the small table 5b where it seems you extrapolate the number of primes of the form x^2+1 which are between 2^n and 2^(n+1). The number of such primes is decreasing as n increases, but it seems to me it should be going up.

C = Number of new primes of the form p=x^2+1 between A and B
The number of primes between 2^n and 2^(n+1) is decreasing,
i round the numbers to integer.

> You wrote in the conclusion,
>
> "The density of the distribution of the primes p=x^2+1 goes surely to zero.
> Supposing that the exploration of the second scenario is rigth, it could be supposed that the primes of the form x^2+1 are limited."
>
> Are you saying that the number of primes of the form x^2+1 appears to be finite?

The density of these primes goes to 0, that does not mean that there could not be any primes of the form x^2+1 any further.
i do not know any algorithm to calculate the evidence of these primes
in advance which might be possible.
Nevertheless the search for those primes might be very difficult.

I would like to compare the results for the polynom x^2+1 with
the results for the polynom 2x^2-1

The last calculation take 23 days, therefore i will need a month
in order to calculate the results for the polynom 2x^2-1
(The Mersenne primes occure on this polynom)

Last but not least i found 191 polynoms
http://109.90.3.58/devalco/basic_polynoms/
which could be examined.

Therefore there is a lot of work :-)
Any help is welcome.

Nice Greetings from the Primes
Bernhard
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