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Re: [PrimeNumbers] Re: Complex a*x^n+b*y^n puzzle

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  • Robdine
    ... From: mikeoakes2 To: primenumbers@yahoogroups.com Sent: Saturday, November 28, 2009 10:09 PM Subject: [PrimeNumbers] Re: Complex a*x^n+b*y^n puzzle Is a
    Message 1 of 37 , Dec 4, 2009
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      ----- Original Message -----
      From: mikeoakes2
      To: primenumbers@yahoogroups.com
      Sent: Saturday, November 28, 2009 10:09 PM
      Subject: [PrimeNumbers] Re: Complex a*x^n+b*y^n puzzle


      Is a gaussian number a+bi prime if a^2+b^2 is a (integer)prime and
      if c^2+d^2 is prime then is also c+di a gaussian prime?

      gr. Rob


      Found by pari-gp script, in a couple of hours' running time:-

      n_max=10
      a=3+6*I, b=7-5*I, x=-2+I, y=-3+I
      n=1 u=-28 + 13*I uu=953
      n=2 u=59 - 76*I uu=9257
      n=3 u=-68 + 293*I uu=90473
      n=4 u=-161 - 926*I uu=883397
      n=5 u=1532 + 2503*I uu=8612033
      n=6 u=-7231 - 5626*I uu=83939237
      n=7 u=27232 + 8813*I uu=819250793
      n=8 u=-89501 + 2374*I uu=8016064877
      n=9 u=262532 - 98777*I uu=78679946753
      n=10 u=-679471 + 559574*I uu=774803901317

      n_max=11
      a=1+8*I, b=7*I, x=-3, y=1-3*I
      n=1 u=18 - 17*I uu=613
      n=2 u=51 + 16*I uu=2857
      n=3 u=-153 - 398*I uu=181813
      n=4 u=-591 + 844*I uu=1061617
      n=5 u=-327 + 268*I uu=178753
      n=6 u=7281 + 8296*I uu=121836577
      n=7 u=11757 - 34688*I uu=1341484393
      n=8 u=-31071 - 6536*I uu=1008126337
      n=9 u=-234387 - 103592*I uu=65668568233
      n=10 u=5961 + 1170376*I uu=1369815514897
      n=11 u=1863717 - 559928*I uu=3786960421273

      n_max=12
      a=26+I, b=16-15*I, x=-2-I, y=-1+I
      n=1 u=-52 + 3*I uu=2713
      n=2 u=44 + 75*I uu=7561
      n=3 u=21 - 286*I uu=82237
      n=4 u=-270 + 677*I uu=531229
      n=5 u=1033 - 1152*I uu=2394193
      n=6 u=-2966 + 1155*I uu=10131181
      n=7 u=6951 + 1024*I uu=49364977
      n=8 u=-13110 - 9503*I uu=262179109
      n=9 u=17453 + 32388*I uu=1353589753
      n=10 u=-3526 - 81765*I uu=6697947901
      n=11 u=-74169 + 165584*I uu=32919101617
      n=12 u=314850 - 254983*I uu=164146852789

      n_max=13
      a=20+12*I, b=27-6*I, x=-2-I, y=1+2*I
      n=1 u=11 + 4*I uu=137
      n=2 u=-45 + 242*I uu=60589
      n=3 u=-217 - 232*I uu=100913
      n=4 u=-761 - 210*I uu=623221
      n=5 u=2131 - 1636*I uu=7217657
      n=6 u=555 - 38*I uu=309469
      n=7 u=7663 + 11248*I uu=185239073
      n=8 u=-18721 - 810*I uu=351131941
      n=9 u=-36709 + 20404*I uu=1763873897
      n=10 u=20355 - 150718*I uu=23130241549
      n=11 u=28343 - 12472*I uu=958876433
      n=12 u=737719 + 142590*I uu=564561231061
      n=13 u=-817949 + 736844*I uu=1211979646937

      n_max=14
      a=20+12*I, b=27-6*I x=-2-I, y=1+2*I
      n=1 u=11 + 4*I uu=137
      n=2 u=-45 + 242*I uu=60589
      n=3 u=-217 - 232*I uu=100913
      n=4 u=-761 - 210*I uu=623221
      n=5 u=2131 - 1636*I uu=7217657
      n=6 u=555 - 38*I uu=309469
      n=7 u=7663 + 11248*I uu=185239073
      n=8 u=-18721 - 810*I uu=351131941
      n=9 u=-36709 + 20404*I uu=1763873897
      n=10 u=20355 - 150718*I uu=23130241549
      n=11 u=28343 - 12472*I uu=958876433
      n=12 u=737719 + 142590*I uu=564561231061
      n=13 u=-817949 + 736844*I uu=1211979646937
      n=14 u=-631845 + 2133802*I uu=4952339079229

      (As before, u = (a*x^n+b*y^n) and uu = norm(u).)

      Mike





      [Non-text portions of this message have been removed]
    • djbroadhurst
      ... There is only rational prime of the form a^2 + b^2 that yields precisely 4 distinct Gaussian primes, namely 2 = 1^2 + 1^2. If a^2 + b^2 is an odd rational
      Message 37 of 37 , Dec 5, 2009
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        --- In primenumbers@yahoogroups.com,
        "Robdine" <robdine@...> wrote:

        > any rational prime that can be represented by the sum
        > of 2 squares (a^2+b^2) will define 4 gaussian primes

        There is only rational prime of the form a^2 + b^2 that yields
        precisely 4 distinct Gaussian primes, namely 2 = 1^2 + 1^2.

        If a^2 + b^2 is an odd rational prime, we have
        8 [sic] asociates of the Gaussian prime z = a + I*b,
        since we may mulitply it and its conjugate z = a - I*b
        by the 4 units I, -I, -1, 1, obtaining 8 distinct
        Gaussian integers that are prime.

        David
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