- 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 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 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

>

> 109.90.3.58/devalco/quadr_Sieb_x^2+1.htm

>

> 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 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

> >

> > 109.90.3.58/devalco/quadr_Sieb_x^2+1.htm

> >

> > 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

> 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 these primes goes to 0, that does not mean that there could not be any primes of the form x^2+1 any further.

>

> "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?

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

http://109.90.3.58/devalco/quadr_Sieb_2x%5E2-1.htm

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