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24833Re: mod quartic composite tests

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  • paulunderwooduk
    Jan 19, 2013
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      --- In primenumbers@yahoogroups.com, "paulunderwooduk" wrote:


      > > >
      > > > {tst(n,x)=kronecker(x^2-4,n)==-1&&
      > > > gcd(x+1,n)==1&&
      > > > Mod(Mod(1,n)*(L+x+1),(L^2-x*L+1)*(L^2+x*L+1))^(n)==-L^3+(x^2-2)*L+x+1;}
      > > >

      Please accept my apology for my previous statements about this composite test. I am actually running tests for:
      (mod n, L^2-x*L+1) and (mod n, L^2+x*L+1); not the product of these. If the main test is passed then gcds with n of x-1, x, x+1, x^2-2 and x^2-3 are logged.

      Now for some speculation about the results so far:

      1) taking the mod with "the product" implies gcd(x,n)==1.

      2) for "the product", gcds of n with x-1, x^2-2 and x^2-3 are either 1 or a prime congruent to 5 (mod 6) where "gcd(x+1,n)" is logged.

      3) logged gcd(x+1,n) is not 1

      4) the logged n are all congruent to 5 (mod 6).

      Paul
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