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Re: mod quartic composite tests

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  • 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 1 of 26 , Jan 11, 2013
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      --- 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 2 of 26 , Jan 15, 2013
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        --- 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 3 of 26 , Jan 15, 2013
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          --- 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 4 of 26 , Jan 15, 2013
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            --- 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 5 of 26 , Jan 17, 2013
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              --- 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 6 of 26 , Jan 17, 2013
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                --- 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 7 of 26 , Jan 17, 2013
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                  --- 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 8 of 26 , Jan 18, 2013
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                    > 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 9 of 26 , Jan 18, 2013
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                      --- 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 10 of 26 , Jan 18, 2013
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                        --- 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 11 of 26 , Jan 18, 2013
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                          --- 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 12 of 26 , Jan 18, 2013
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                            --- 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 13 of 26 , Jan 19, 2013
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                              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 14 of 26 , Jan 19, 2013
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                                --- 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 15 of 26 , 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
                                • 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 16 of 26 , 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
                                  • 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 17 of 26 , Jan 21, 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;}
                                      > > > >
                                      >

                                      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 18 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 19 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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