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Re: mod quartic composite tests

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  • paulunderwooduk
    ... I have verified this test for all x for all n
    Message 1 of 26 , Jan 10 10:28 AM
      --- In primenumbers@yahoogroups.com, "paulunderwooduk" wrote:

      > gcd(x^3-x,n)==1
      > kronecker(x^2-4,n)==-1
      > gcd(x^2-2,n)==1
      > (L+x^2-2)^n==-L^3+(x^2-2)*L+(x^2-2) (mod n, (L^2-x*L+1)*(L^2+x*L+1))

      I have verified this test for all x for all n<10^6 co-prime to 30,

      Paul
    • paulunderwooduk
      ... ? {tst(n,x)=kronecker(x^2-4,n)==-1&& gcd(x^3-x,n)==1&& gcd(x^2-2,n)==1&& Mod(Mod(1,n)*(L+x^2-2),(L^2-x*L+1)*(L^2+x*L+1))^(n)==-L^3+(x^2-2)*L+x^2-2;} ?
      Message 2 of 26 , Jan 11 12:17 PM
        --- In primenumbers@yahoogroups.com, "paulunderwooduk" wrote:
        >
        >
        >
        > --- In primenumbers@yahoogroups.com, "paulunderwooduk" wrote:
        >
        > > gcd(x^3-x,n)==1
        > > kronecker(x^2-4,n)==-1
        > > gcd(x^2-2,n)==1
        > > (L+x^2-2)^n==-L^3+(x^2-2)*L+(x^2-2) (mod n, (L^2-x*L+1)*(L^2+x*L+1))
        >
        > I have verified this test for all x for all n<10^6 co-prime to 30,
        >

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

        ? {tstfile("underw297.txt");}
        2/297 counterexamples left in underw297.txt
        ? {tstfile("underw65.txt");}
        5146/12846 counterexamples left in underw65.txt

        Paul
      • paulunderwooduk
        ... Adding gcd(x^2-3,n)==1 makes sense because x^2-2==1 (mod x^2-3). So the resurrected test is: {tst(n,x)=kronecker(x^2-4,n)==-1&& gcd(x^3-x,n)==1&&
        Message 3 of 26 , Jan 11 6:11 PM
          --- In primenumbers@yahoogroups.com, "paulunderwooduk" wrote:

          >
          > ? {tst(n,x)=kronecker(x^2-4,n)==-1&&
          > gcd(x^3-x,n)==1&&
          > gcd(x^2-2,n)==1&&
          > Mod(Mod(1,n)*(L+x^2-2),(L^2-x*L+1)*(L^2+x*L+1))^(n)==-L^3+(x^2-2)*L+x^2-2;}
          >
          > ? {tstfile("underw297.txt");}
          > 2/297 counterexamples left in underw297.txt
          > ? {tstfile("underw65.txt");}
          > 5146/12846 counterexamples left in underw65.txt
          >

          Adding gcd(x^2-3,n)==1 makes sense because x^2-2==1 (mod x^2-3). So the resurrected test is:

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

          Then checking David's files at:
          http://physics.open.ac.uk/~dbroadhu/cert/

          {tstfile(file)=local(c,n,x,v=readvec(file));
          for(k=1,#v,n=v[k][1];x=v[k][2];
          if(tst(n,x)&&!isprime(n),c++));
          print(c"/"#v" counterexamples left in "file);c;}

          ? {tstfile("underbh4.txt");}
          0/33445 counterexamples left in underbh4.txt
          ? {tstfile("underbh6.txt");}
          0/308619 counterexamples left in underbh6.txt
          ? {tstfile("underw97.txt");}
          0/97 counterexamples left in underw97.txt
          ? {tstfile("underw297.txt");}
          0/297 counterexamples left in underw297.txt
          ? {tstfile("underw65.txt");}
          0/12846 counterexamples left in underw65.txt
          ? {tstfile("underw65x.txt");}
          0/10220 counterexamples left in underw65x.txt
          ? {tstfile("underwg.txt");}
          0/100000 counterexamples left in underwg.txt

          Paul
        • paulunderwooduk
          ... n=2672279 and x=89805 forms a near-counterexample, saved only by gcd(x^3-x,n)==2672279. Since gcd(x^3-x,n)==1 is required I do not have to verify numbers
          Message 4 of 26 , Jan 15 12:31 PM
            --- In primenumbers@yahoogroups.com, "paulunderwooduk" wrote:

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

            n=2672279 and x=89805 forms a near-counterexample, saved only by gcd(x^3-x,n)==2672279.

            Since gcd(x^3-x,n)==1 is required I do not have to verify numbers divisible by 2 or 3, and with kronecker(x^2-4,n)==-1, I do not have to verify numbers divisible by 5. Also the squares modulo 7 are 0, 1, 4, and 2, and because I am checking gcd(x^3-x,n)==1, kronecker(x^2-4,n)==-1 and gcd(x^2-2,n)==1, I can skip numbers divisible by 7. All in all, I need only verify numbers co-prime to 210,

            Paul
          • paulunderwooduk
            ... That should read gcd(x^2-3,n)==2672279. Paul
            Message 5 of 26 , Jan 15 12:35 PM
              --- In primenumbers@yahoogroups.com, "paulunderwooduk" wrote:

              > n=2672279 and x=89805 forms a near-counterexample, saved only by gcd(x^3-x,n)==2672279.

              That should read gcd(x^2-3,n)==2672279.

              Paul
            • djbroadhurst
              ... A wise wriggle :-) Without it, you left yourself wide open to fraud. With it, you seem to be much more secure. I believe, yet cannot show, that there are
              Message 6 of 26 , Jan 15 2:30 PM
                --- In primenumbers@yahoogroups.com,
                "paulunderwooduk" wrote:

                > Adding gcd(x^2-3,n)==1

                A wise "wriggle" :-)

                Without it, you left yourself wide open to fraud.
                With it, you seem to be much more secure.

                I believe, yet cannot show, that there are zillions of
                counterexamples to your single-parameter, double-Frobenius
                test, yet the prospects of finding one, now that you
                have bolted the stable door, seem to be minuscule.

                Thanks, Paul, for your responsiveness.

                David
              • djbroadhurst
                ... http://physics.open.ac.uk/~dbroadhu/cert/underwqd.txt.gz provides 352869 such frauds: {tst(n,x)=kronecker(x^2-4,n)==-1&&gcd(x^3-x,n)==1&&gcd(x^2-2,n)==1&&
                Message 7 of 26 , Jan 17 5:28 AM
                  --- In primenumbers@yahoogroups.com,
                  "paulunderwooduk" wrote:

                  > n=2672279 and x=89805

                  http://physics.open.ac.uk/~dbroadhu/cert/underwqd.txt.gz
                  provides 352869 such frauds:

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

                  {tstfile(file)=local(v=readvec(file),c);
                  c=sum(k=1,#v,tst(v[k][1],v[k][2])&&!isprime(v[k][1]));
                  print(c"/"#v" counterexamples left in "file);c;}

                  tstfile("underwqd.txt");

                  352869/352869 counterexamples left in underwqd.txt

                  All are trapped by Paul's latest wriggle,
                  which requires x^2-3 to be coprime to n.

                  David
                • paulunderwooduk
                  ... Thanks for these, David. Is it a comprehensive list for all n
                  Message 8 of 26 , Jan 17 9:29 AM
                    --- In primenumbers@yahoogroups.com, "djbroadhurst" wrote:

                    > http://physics.open.ac.uk/~dbroadhu/cert/underwqd.txt.gz

                    >
                    > 352869/352869 counterexamples left in underwqd.txt
                    >
                    > All are trapped by Paul's latest wriggle,
                    > which requires x^2-3 to be coprime to n.
                    >

                    Thanks for these, David. Is it a comprehensive list for all n <= 97847746461047271599?

                    Paul
                  • djbroadhurst
                    ... By no means. However the updated file http://physics.open.ac.uk/~dbroadhu/cert/underwqd.txt.gz now has more:
                    Message 9 of 26 , Jan 17 1:23 PM
                      --- In primenumbers@yahoogroups.com,
                      "paulunderwooduk" wrote:

                      > Is it a comprehensive list for all n <= 97847746461047271599?

                      By no means. However the updated file
                      http://physics.open.ac.uk/~dbroadhu/cert/underwqd.txt.gz
                      now has more:

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

                      {tstfile(file)=local(v=readvec(file),c);
                      c=sum(k=1,#v,tst(v[k][1],v[k][2])&&!isprime(v[k][1]));
                      print(c"/"#v" counterexamples left in "file);c;}

                      tstfile("underwqd.txt");

                      422355/422355 counterexamples left in underwqd.txt

                      Challenge: Find a composite 10^10-smooth positive
                      integer, n, such that:
                      1) there exist an integer x that passes tst(n,x),
                      2) n is not in underwqd.txt.

                      Comment: I do not know of any such integer. Nor do I know
                      how to search for one. So I guess that means my gremlins
                      are comprehensively exhausted, though the question is not.

                      David
                    • djbroadhurst
                      ... Exercise: Show that Paul s test Mod(Mod(1,n)*(L+x^2-2),(L^2-x*L+1)*(L^2+x*L+1))^n==-L^3+(x^2-2)*L+x^2-2 requires n to be a Fermat pseudoprime in base
                      Message 10 of 26 , Jan 18 12:47 AM
                        > All are trapped by Paul's latest wriggle,
                        > which requires x^2-3 to be coprime to n.

                        Exercise: Show that Paul's test
                        Mod(Mod(1,n)*(L+x^2-2),(L^2-x*L+1)*(L^2+x*L+1))^n==-L^3+(x^2-2)*L+x^2-2
                        requires n to be a Fermat pseudoprime in base 1+(x^2-3)*(x^2-2)^3
                        and thus loses (at least) one selfridge of potency for x^2 = 3 mod n.

                        David
                      • paulunderwooduk
                        ... ? M=[0,(x^2-2),0,-1;1,0,0,0;0,1,0,0;0,0,1,0];matdet(M+x^2-2)==1+(x^2-3)*(x^2-2)^3 1 Thanks for the insight. This can be split into 2 Fermat tests: ?
                        Message 11 of 26 , Jan 18 1:23 AM
                          --- In primenumbers@yahoogroups.com, "djbroadhurst" wrote:
                          >
                          >
                          >
                          > > All are trapped by Paul's latest wriggle,
                          > > which requires x^2-3 to be coprime to n.
                          >
                          > Exercise: Show that Paul's test
                          > Mod(Mod(1,n)*(L+x^2-2),(L^2-x*L+1)*(L^2+x*L+1))^n==-L^3+(x^2-2)*L+x^2-2
                          > requires n to be a Fermat pseudoprime in base 1+(x^2-3)*(x^2-2)^3
                          > and thus loses (at least) one selfridge of potency for x^2 = 3 mod n.
                          >

                          ? M=[0,(x^2-2),0,-1;1,0,0,0;0,1,0,0;0,0,1,0];matdet(M+x^2-2)==1+(x^2-3)*(x^2-2)^3
                          1

                          Thanks for the insight.

                          This can be split into 2 Fermat tests:

                          ? M=[x,-1;1,0];matdet(M+x^2-2)
                          x^4 + x^3 - 4*x^2 - 2*x + 5
                          ? M=[-x,-1;1,0];matdet(M+x^2-2)
                          x^4 - x^3 - 4*x^2 + 2*x + 5

                          which are equal to:

                          ? M=[x,-1;1,0];matdet(M+x^2-2)==(x^2-2)*(x+2)*(x-1)+1
                          1
                          ? M=[-x,-1;1,0];matdet(M+x^2-2)==(x^2-2)*(x-2)*(x+1)+1
                          1

                          Paul
                        • djbroadhurst
                          ... Exercise 2: Show that the test loses 3 selfrides for x^2 = 3 mod n. Comment 2: Hence the happy gremlins, in this case. Exercise 3: Show that the test loses
                          Message 12 of 26 , Jan 18 3:56 AM
                            --- In primenumbers@yahoogroups.com, "paulunderwooduk" wrote:

                            > > loses (at least) one selfridge of potency for x^2 = 3 mod n.
                            > Thanks for the insight.

                            Exercise 2: Show that the test loses 3 selfrides for x^2 = 3 mod n.
                            Comment 2: Hence the happy gremlins, in this case.

                            Exercise 3: Show that the test loses 1 selfride for 2*x^2 = 5 mod n.
                            Comment 3: The gremlins were not able to fool it in this case.

                            David
                          • djbroadhurst
                            ... Exercise 4: Show that the test loses 2 selfridges for 2*x^2 = 5 mod n. David
                            Message 13 of 26 , Jan 18 4:06 AM
                              --- In primenumbers@yahoogroups.com, "djbroadhurst" wrote:
                              > Exercise 3: Show that the test loses 1 selfridge for 2*x^2 = 5 mod n.

                              Exercise 4: Show that the test loses 2 selfridges for 2*x^2 = 5 mod n.

                              David
                            • djbroadhurst
                              ... Exercise 5: Show that the test loses 3 selfridges for 2*x^2 = 5 mod n, degenerating to a 1-selfridge Euler test, with base -15/16, plus a 2-selfridge Lucas
                              Message 14 of 26 , Jan 18 4:43 AM
                                --- In primenumbers@yahoogroups.com, "djbroadhurst" wrote:

                                > Exercise 4: Show that the test loses 2 selfridges for 2*x^2 = 5 mod n.

                                Exercise 5: Show that the test loses 3 selfridges for 2*x^2 = 5 mod n,
                                degenerating to a 1-selfridge Euler test, with base -15/16, plus a
                                2-selfridge Lucas test with P = 2/5 and Q = 1, and thus costs the same as BPSW.

                                Comment: As in the case of BPSW, the gremlins cannot defraud this case.

                                David
                              • paulunderwooduk
                                I failed to solve any of David s exercises... but I have done some shallow verification, n
                                Message 15 of 26 , Jan 19 8:35 AM
                                  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,

                                  Paul -- subject to gcd wriggles
                                • paulunderwooduk
                                  ... Further, it seems that if gcd(x+1,n) is needed then it is equal to 1 (mod 6) Paul
                                  Message 16 of 26 , Jan 19 9:22 AM
                                    --- 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)

                                    Paul
                                  • paulunderwooduk
                                    ... 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
                                    Message 17 of 26 , Jan 19 9:44 AM
                                      --- 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
                                    • paulunderwooduk
                                      ... 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);
                                      Message 18 of 26 , Jan 19 1:05 PM
                                        --- 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
                                      • paulunderwooduk
                                        ... 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&&
                                        Message 19 of 26 , Jan 21 3:27 AM
                                          --- 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;}
                                          > > > >
                                          >

                                          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;}

                                          Paul
                                        • 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 20 of 26 , Jan 27 2:23 AM
                                            --- 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 21 of 26 , Jan 30 1:18 PM
                                              > --- 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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