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Re: 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
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      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 integer n >= 1.
      If k = 0 to 3, these can be summarized as:-
      Gk(n) = n*(1+i)^n + (i)^k.
      G0 and G2 bear a close resemblence to the Cullen
      C(n) = n*2^n + 1
      and Woodall
      W(n) = n*2^n - 1
      numbers, respectively; the other 2 have no such rational-integer counterparts.
      All 4 are quite similar to their rational counterparts in respect of
      factorability.

      Because (1+i)^2 = 2*i, (1+i)^4 = -4, (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.
      In all other cases, Gk(n) is complex, and we work with its modulus:
      Mk(n) = norm(Gk(n)) =
    • 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 2 of 2 , Dec 29, 2000
      • 0 Attachment
        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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