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24832Re: mod quartic composite tests

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  • paulunderwooduk
    Jan 19, 2013
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      --- In primenumbers@yahoogroups.com, "paulunderwooduk" wrote:
      >
      >
      >
      > --- In primenumbers@yahoogroups.com, "paulunderwooduk" wrote:
      > >
      > >
      > > I failed to solve any of David's exercises... but I have done some shallow verification, n<2.5*10^5, for yet another test:
      > >
      > > {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;}
      > >
      > > It looks as though when "gcd(x+1,n)==1" is needed then n==5 (mod 6), and any of gcd(x-1,n), gcd(x,n), gcd(x^2-2,n) and gcd(x^3-3,n) greater than 1 is also a prime equal to 5 (mod 6) when "gcd(x+1,n)==1" is required,
      > >
      >
      > Further, it seems that if "gcd(x+1,n)" is needed then it is equal to 1 (mod 6)
      >

      These ancillary statements are mostly false, except that maybe when "gcd(x+1,n)" needs to be checked then gcd(x-1,n), gcd(x,n), gcd(x^2-2,n) and gcd(x^3-3,n) are either 1 or prime,

      Paul
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