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Gaussian analogues of the Cullen and Woodall primes

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  • mikeoakes2@aol.com
    Over the last 6 months I have been investigating the following 4 inter-related Gaussian integer sequences:- G0(n) = n*(1+i)^n + 1 G1(n) = n*(1+i)^n + i G2(n) =
    Message 1 of 2 , Dec 29, 2000
      Over the last 6 months I have been investigating the following 4
      inter-related Gaussian integer sequences:-
      G0(n) = n*(1+i)^n + 1
      G1(n) = n*(1+i)^n + i
      G2(n) = n*(1+i)^n - 1
      G3(n) = n*(1+i)^n - i
      for rational integer n >= 1.
      Letting k = 0 to 3, these can be condensed as:-
      Gk(n) = n*(1+i)^n + (i)^k.

      G0 and G2 bear a close resemblance to the Cullen
      C(n) = n*2^n + 1
      and Woodall
      W(n) = n*2^n - 1
      numbers, respectively; and all 4 are quite similar to these real counterparts
      in respect of factorability.

      Because (1+i)^8 = 16, the form taken by the various Gk(n) depends only on the
      value of (n mod 8).

      For n = 0 or 4 mod 8, G1(n) and G2(n) are real.
      For n = 2 or 6 mod 8, G3(n) and G4(n) are pure imaginary.
      Such Gk(n) give rise to the following 2 rational-integer sequences:-
      Gp(n) = n*2^(n/2) + 1
      Gm(n) = n*2^(n/2) - 1
      where n is even.
      Gp(n) is divisible by 3 if n = 8 or 10 mod 12; is divisible by 5 if n = 2,
      24, 36 or 38 mod 40; and is prime for the following n:-
      2 2*2^1 + 1
      4 4*2^2 + 1
      12 12*2^6 + 1
      52 52*2^26 + 1
      100 100*2^50 + 1
      108 108*2^54 + 1
      160 160*2^80 + 1
      2940 2940*2^1470 + 1
      2964 2964*2^1482 + 1
      17334 17334*2^8667 + 1
      21768 21768*2^10884 + 1
      41604 41604*2^20802 + 1
      65208 65208*2^32604 + 1
      72780 72780*2^36390 + 1
      Gn(n) is divisible by 3 if n = 2 or 4 mod 12; is divisible by 5 if n = 4, 66,
      18 or 22 mod 40; and is prime for the following n:-
      2 2*2^1 - 1
      6 6*2^3 - 1
      8 8*2^4 - 1
      20 20*2^10 - 1
      54 54*2^27 - 1
      68 68*2^34 - 1
      468 468*2^234 - 1
      648 648*2^324 - 1
      1100 1100*2^550 - 1
      1374 1374*2^687 - 1
      14072 14072*2^7036 - 1
      17790 17790*2^8895 - 1
      20038 20038*2^10019 - 1
      27192 27192*2^13596 - 1
      42692 42692*2^21346 - 1

      In all other cases, the Gk(n) are complex, and we work with the norm:
      Nk(n) = [n*(1+i)^n + (i)^k] * [n*(1-i)^n + (-i)^k]
      Letting n and k vary, these Nk(n) give rise to the following 3
      rational-integer sequences:-
      Ne(n) = n^2*2^n + 1
      where n is even,
      and
      Nop(n) = n^2*2^n + n*2^((n+1)/2) + 1
      Nom(n) = n^2*2^n - n*2^((n+1)/2) + 1
      where n is odd.

      Ne(n) is divisible by 5 if n = 6, 8, 12 or 14 mod 20; is never divisible by
      any prime which is = 7 mod 8; and is prime for the following n:-
      2 2^2*2^2 + 1
      4 4^2*2^4 + 1
      30 30^2*2^30 + 1
      100 100^2*2^100 + 1
      142 142^2*2^142 + 1
      144 144^2*2^144 + 1
      150 150^2*2^150 + 1
      198 198^2*2^198 + 1
      304 304^2*2^304 + 1
      782 782^2*2^782 + 1
      858 858^2*2^858 + 1
      3638 3638^2*2^3638 + 1
      6076 6076^2*2^6076 + 1
      12876 12876^2*2^12876 + 1
      30180 30180^2*2^30180 + 1
      48470 48470^2*2^48470 + 1
      Nop(n) is divisible by 5 if n = 1, 3, 7, 19, 29, 31, 33 or 37 mod 40; and is
      prime for the following n:-
      1 1^2*2^1 + 1*2^1 + 1
      9 9^2*2^9 + 9*2^5 + 1
      61 61^2*2^61 + 61*2^31 + 1
      143 143^2*2^143+143*2^72 + 1
      159 159^2*2^159+159*2^80 + 1
      387 387^2*2^387+387*2^194 + 1
      1137 1137^2*2^1137+1137*2^569 + 1
      1973 1973^2*2^1973+1973*2^987 + 1
      3337 3337^2*2^3337+3337*2^1669 + 1
      16895 16895^2*2^16895+16895*2^8448 + 1
      37171 37171^2*2^37171+37171*2^18586 + 1
      Nom(n) is divisible by 5 if n = 9, 11, 13, 17, 21, 23, 27 or 39 mod 40; and
      is prime for the following n:-
      3 3^2*2^3 - 3*2^2 + 1
      5 5^2*2^5 - 5*2^3 + 1
      19 19^2*2^19 - 19*2^10 + 1
      25 25^2*2^25 - 25*2^13 + 1
      29 29^2*2^29 - 29*2^15 + 1
      47 47^2*2^47 - 47*2^24 + 1
      167 167^2*2^167 - 167*2^84 + 1
      407 407^2*2^407 - 407*2^204 + 1
      3909 3909^2*2^3909 - 3909*2^1955 + 1
      4433 4433^2*2^4433 - 4433*2^2217 + 1
      4845 4845^2*2^4845 - 4845*2^2423 + 1
      4921 4921^2*2^4921 - 4921*2^2461 + 1
      30349 30349^2*2^30349 - 30349*2^15175 + 1
      78873 74873^2*2^74873 - 74873*2^37437 + 1

      That is the complete prime picture for n up to 100,000. Chris Nash's fine
      PFGW program was used, both in the search and to prove primality, as all
      these forms are amenable to the deterministic primality tests developed by
      Pocklington, Brillhart-Lehmer-Selfridge and others.

      Only the very last of these makes Chris Caldwell's top 5000 database, but the
      search is continuing, for n > 100,000. Any volunteers??

      Mike Oakes
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