limes resp. prime distribution for f(n)=n²+1

[bhelmes]

A peaceful morning for you,


if you regard the quotient between

the primes p with p=f(n)=n²+1 and

the primes with p | f(n) appearing the second times with increasing n
the limes seems to go to 1


I calculate this sequence up to n=2^39
(Results under http://devalco.de/quadr_Sieb_x%5E2+1.php#4g

and graphic)


Is this a conjecture or are there any similiar proofs ?


Greeting from the gaussian primes

Bernhard

[CRGreathouse]

I can't tell what you're saying. At first I thought you were comparing
A002496, the primes of the form n^2+1
to
A002313, the primes dividing n^2+1 for some n
but I could find only evidence of the former in your numbers (the orange column).

You clearly don't mean anything close to the usual definition of "all primes", and "appearing the second times" is quite opaque.

[Dr Sardonicus]

I'm not sure what "limes" means here; lim sup perhaps?

If p == 1 (mod 4), there are exactly two integers 0 < n₁ < n₂ < p, with

n₂ = p - n₁, for which

n₁^2 == n₂^2 == -1 (mod p).

If n₁/p = k, then n₂/p = 1 - k.

I am guessing it is known that lim inf k = 0, so lim sup 1 - k = 1.

[bhelmes]

Thanks for your reply,



i try to improve my mathematical precision expressions:


if you regard the quotient between

A) the amount of primes p with p=f(n)=n²+1 (A002496) and

B) the amount of primes with p | f(n)

appearing the second times with increasing n

[5|f(3), 13|f(8), 17|f(13), 29|f(17), 37|f(31)...]
( i did not find this sequence in OEIS )



the limes for n-> oo seems to go to 1


I calculate this sequence up to n=2^39
(Results under http://devalco.de/quadr_Sieb_x%5E2+1.php#4g and graphic)

Is it possible to proof the limes mathematical correctly ?
The sequence is monoton declining up from n=2^17
and i can proof that the amount for B) is infinitiv



Greeting from the gaussian primes

Bernhard

[VBCurtis]

What does "limes" mean? I thought it was a typo the first time you used it, but you used it again so now I think it's intentional.

[bhelmes]

Originally Posted by VBCurtis

What does "limes" mean? I thought it was a typo the first time you used it, but you used it again so now I think it's intentional.

I thought limes is the latin word for limit and is used in the mathematical area.

[a1call]

It is the Latin word for limit according to google translate.

But I can't find any reference in English to use it as limit other than the fact that it is the root of the word limit.
Below is everything you never wanted to know about the Latin limes.


https://en.m.wikipedia.org/wiki/Limes

ETA It can also mean limit in German.

[CRGreathouse]

Originally Posted by bhelmes

A) the amount of primes p with p=f(n)=n²+1 (A002496) and

B) the amount of primes with p | f(n)

appearing the second times with increasing n

[5|f(3), 13|f(8), 17|f(13), 29|f(17), 37|f(31)...]
( i did not find this sequence in OEIS )

What do you mean by "appearing the second times with increasing n"?

Because again, the primes p dividing f(n) = n^2+1 for some n are just A002313. So you're excluding 2 and 29 and lots of other values for some reason I don't understand.

[bhelmes]

Originally Posted by CRGreathouse

What do you mean by "appearing the second times with increasing n"?

p | f(n) with n > (p-n), n < p

Originally Posted by CRGreathouse

Because again, the primes p dividing f(n) = n^2+1 for some n are just A002313. So you're excluding 2 and 29 and lots of other values for some reason I don't understand.

The 2 may be a special case which can be negligable if you consider the limes.

29 | f(12) (1.) and 29 | f(17) (2.)
I consider the (2.) case.


The number of the sequence A00231 are identical but the order / arrangement is different.



Actually i do a sieve construction. You could sieve the primes p | f(n)
either for the first time or for the second time.


I have added two very simple sourcecode in C which are very easy in order to understand the construction:


http://devalco.de/quadr_Sieb_x%5E2+1.php see point 3. j) k)


Thanks a lot for your patience

Bernhard

[bhelmes]

If the romulus interpreter from Thailand is not present,
my expressions are poor and low.

[Batalov]

Originally Posted by bhelmes

...can be negligable if you consider the limes.

Originally Posted by bhelmes

If the romulus interpreter from Thailand is not present,
my expressions are poor and low.

My bet is he would interpret:

"She put lime in the coconut,
She drank them both up..."