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Re: mod quartic composite tests

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  • paulunderwooduk
    ... n=2953711;x=285843 is a near-counterexample that comes from me testing over the two quadratics that form the quartic. gcd(x,n)==95281 and
    Message 1 of 26 , Jan 27, 2013
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      --- In primenumbers@yahoogroups.com, "paulunderwooduk" wrote:

      > Here is another test, on the same theme, for which I cannot also easily find a fraud:
      >
      > {tst(n,x)=kronecker(x^2-4,n)==-1&&
      > gcd(x^2-1,n)==1&&
      > Mod(Mod(1,n)*(L+x^2-1),(L^2-x*L+1)*(L^2+x*L+1))^(n)==-L^3+(x^2-2)*L+x^2-1;}
      >

      n=2953711;x=285843 is a near-counterexample that comes from me testing over the two quadratics that form the quartic. gcd(x,n)==95281 and gcd(x^2-2,n)==31,

      Paul
    • paulunderwooduk
      ... n=1934765;x=1219266 has gcd(x-1,n)==265. So 2) might be 1 or a product of primes each congruent 5 (mod 6) , but as greater n get tested I guess this rule
      Message 2 of 26 , Jan 30, 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.
        >

        n=1934765;x=1219266 has gcd(x-1,n)==265. So 2) might be "1 or a product of primes each congruent 5 (mod 6)", but as greater n get tested I guess this rule will break too...

        > 3) logged gcd(x+1,n) is not 1
        >
        > 4) the logged n are all congruent to 5 (mod 6).
        >

        I have verified all n<1.95*10^6

        Paul
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