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24305Re: Prime sieving on the polynomial f(n)=n^2+1

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  • bhelmes_1
    Jul 7, 2012
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      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
      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
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