Loading ...
Sorry, an error occurred while loading the content.

Re: primes of the form x^3 - y^2

Expand Messages
  • David Broadhurst
    ... for primes p such that x^3 - p is conjecturally never a square for integer x. To prove that there are no integral points on the elliptic curves x^3 = y^2 +
    Message 1 of 22 , Jun 17, 2009
    View Source
    • 0 Attachment
      --- In primenumbers@yahoogroups.com,
      cino hilliard <hillcino368@...> wrote:

      > 3, 5, 17, 29, 31, 37, 41, 43, 59, 73, 97,
      > 101, 103, 113, 131, 137, 149, 157, 163 ...

      for primes p such that x^3 - p is conjecturally never
      a square for integer x.

      To prove that there are no integral points on the
      elliptic curves x^3 = y^2 + p, for those primes,
      you might need to compute their Mordell-Weil groups:
      http://www.springerlink.com/content/p80721u575l12047/
      www.dpmms.cam.ac.uk/Algebraic-Number-Theory/0086/paper.ps

      There are 21 integral points with y > 0 on the elliptic curve
      x^3 = y^2 + 28279
      with a Mordell-Weil group of rank 5, namely those with
      x = 32, 34, 40, 50, 67, 70, 122, 260, 295, 359, 515, 592,
      952, 2284, 2327, 2410, 7330, 7580, 11702, 130184, 26507590.

      David
    • David Broadhurst
      ... http://magma.maths.usyd.edu.au/calc/ enabled me to show that there are no more: Input: E := EllipticCurve([0, -28279]); Q, reps := IntegralPoints(E); Q;
      Message 2 of 22 , Jun 17, 2009
      View Source
      • 0 Attachment
        --- In primenumbers@yahoogroups.com,
        "David Broadhurst" <d.broadhurst@...> wrote:

        > There are 21 integral points with y > 0 on the elliptic curve
        > x^3 = y^2 + 28279
        > with a Mordell-Weil group of rank 5, namely those with
        > x = 32, 34, 40, 50, 67, 70, 122, 260, 295, 359, 515, 592,
        > 952, 2284, 2327, 2410, 7330, 7580, 11702, 130184, 26507590.

        http://magma.maths.usyd.edu.au/calc/
        enabled me to show that there are no more:

        Input:

        E := EllipticCurve([0, -28279]);
        Q, reps := IntegralPoints(E);
        Q;

        Output:

        [ (34 : -105 : 1), (40 : 189 : 1), (70 : 561 : 1),
        (32 : 67 : 1), (50 : -311 : 1), (67 : -522 : 1),
        (122 : 1337 : 1), (260 : -4189 : 1), (295 : 5064 : 1),
        (592 : -14403 : 1), (952 : -29373 : 1), (359 : 6800 : 1),
        (515 : 11686 : 1), (2284 : 109155 : 1), (2410 : 118311 : 1),
        (2327 : -112252 : 1), (7330 : 627561 : 1), (7580 : -659939 : 1),
        (11702 : -1265873 : 1), (130184 : -46971715 : 1),
        (26507590 : 136475711439 : 1) ]

        The CPU-time was comfortably below the 20 second limit:

        > Total time: 7.219 seconds, Total memory usage: 137.04MB

        David
      • David Broadhurst
        ... E := EllipticCurve([0, -101]); Q, reps := IntegralPoints(E); Q; [] Total time: 0.370 seconds, Total memory usage: 137.04MB But for x^3-y^2=149, Magma
        Message 3 of 22 , Jun 17, 2009
        View Source
        • 0 Attachment
          --- In primenumbers@yahoogroups.com,
          cino hilliard <hillcino368@...> wrote:

          > My gut is x^3-y^2 = 101,103 and others are out there but we
          > will not have the time to find them in this BB universe.

          E := EllipticCurve([0, -101]);
          Q, reps := IntegralPoints(E);
          Q;

          []
          Total time: 0.370 seconds, Total memory usage: 137.04MB

          But for x^3-y^2=149, Magma issues a warning:

          E := EllipticCurve([0, -149]);
          Q, reps := IntegralPoints(E);
          Q;

          Warning: rank computed (0) is only a lower bound
          (It may still be correct, though)
          []
          Total time: 0.340 seconds, Total memory usage: 61.95MB

          PS: It seems that Sm*r*nd*ch* once conjectured that there
          are no rational points on x^3 = y^2 + 7, somehow
          overlooking the fact that 2^3 = 1^2 + 7:
          www.articlearchives.com/asia/northern-asia-china-south/951072-1.html

          David
        • David Broadhurst
          Off list, Tony Noe kindly provided this link: http://www.research.att.com/~njas/sequences/A081120 From the table attached thereto, one may conclude that Cino s
          Message 4 of 22 , Jun 18, 2009
          View Source
          • 0 Attachment
            Off list, Tony Noe kindly provided this link:
            http://www.research.att.com/~njas/sequences/A081120

            From the table attached thereto, one may conclude that
            Cino's sequence, "Primes not of the form x^3 - y^2", begins

            3, 5, 17, 29, 31, 37, 41, 43, 59, 73, 97, 101, 103, 113, 131,
            137, 149, 157, 163, 173, 179, 181, 197, 211, 227, 229, 241,
            257, 263, 269, 281, 283, 311, 313, 317, 331, 337, 347, 349,
            353, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 443,
            449, 457, 461, 467, 479, 491, 509, 521, 523, 541, 563, 569,
            571, 577, 601, 607, 613, 617, 619, 641, 643, 653, 659, 661,
            677, 691, 701, 709, 733, 739, 743, 751, 757, 761, 773, 787,
            809, 811, 821, 823, 827, 829, 839, 853, 857, 863, 877, 881,
            883, 907, 911, 929, 937, 941, 947, 953, 967, 977, 983, 997 ...

            This paper appears to have open access:
            http://tinyurl.com/mtvsn5
            See Section 5.1 for Cino's record holder, p = 28279,
            with 21 integral points.

            David
          • cino hilliard
            Hi David, The elliptic functions in Pari are forboding at least to me. I contacted the Pari group and got a workaround for eliptic curves from Karim. He gave
            Message 5 of 22 , Jun 18, 2009
            View Source
            • 0 Attachment
              Hi David,

              The elliptic functions in Pari are forboding at least to me.

              I contacted the Pari group and got a workaround for eliptic curves from
              Karim. He gave the following to find solutions for x^3-y^2 = p.



              diffcubes(n,p)=local(x,y);setintersect(vector(n,x,x^3-p),vector(n,y,y^2))
              which we found out will work in the next version of Pari. :-)



              So Karim wrote:

              > For "older" versions than that, use
              >
              > setintersect(Set(vector(n,x,x^3-p)), Set(vector(n,y,y^2)))
              >
              > ( slower but not *much* slower... )


              Ok, I did it my way with this and asked a question at the end.
              The timing aint bad. Magma and Sage would probably be faster offline.

              We need a way of knowing upper bounds a priori though.


              > diffcubes(n,p) =
              > {
              > local(j,x,y,c);
              > a=eval(setintersect(Set(vector(n,x,x^3-p)), Set(vector(n,y,y^2))));
              > c=length(a);
              > a=vecsort(a);
              > for(j=1,c,
              > y=round(sqrt(a[j])); \\ this could be iffy
              > x=round((a[j]+p)^(1/3));
              > print(j": "x"^3 - "y"^2 = "p); \\ Too fancy? Change it.
              > );
              > c;
              > }
              >
              >
              >(14:51:50) gp > diffcubes(300000,431)
              >1: 8^3 - 9^2 = 431
              >2: 11^3 - 30^2 = 431
              >3: 20^3 - 87^2 = 431
              >4: 30^3 - 163^2 = 431
              >5: 36^3 - 215^2 = 431
              >6: 138^3 - 1621^2 = 431
              >7: 150^3 - 1837^2 = 431
              >8: 575^3 - 13788^2 = 431
              >9: 3903^3 - 243836^2 = 431
              >(15:02:35) gp > ##
              > *** last result computed in 2,250 ms.
              >(15:02:39) gp >
              >
              >This finds all instances because I new a prori 243836 was the big kahuna.
              >I guess it will still be trial and error?
              >
              >Oh well, we are at least 2 quanta over what I had.
              >
              > Thank you >>>
              >Cino


              Thanks for all your work,

              Cino



              To: primenumbers@yahoogroups.com
              From: d.broadhurst@...
              Date: Thu, 18 Jun 2009 16:26:36 +0000
              Subject: [PrimeNumbers] Re: primes of the form x^3 - y^2







              Off list, Tony Noe kindly provided this link:
              http://www.research.att.com/~njas/sequences/A081120

              From the table attached thereto, one may conclude that
              Cino's sequence, "Primes not of the form x^3 - y^2", begins

              3, 5, 17, 29, 31, 37, 41, 43, 59, 73, 97, 101, 103, 113, 131,
              137, 149, 157, 163, 173, 179, 181, 197, 211, 227, 229, 241,
              257, 263, 269, 281, 283, 311, 313, 317, 331, 337, 347, 349,
              353, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 443,
              449, 457, 461, 467, 479, 491, 509, 521, 523, 541, 563, 569,
              571, 577, 601, 607, 613, 617, 619, 641, 643, 653, 659, 661,
              677, 691, 701, 709, 733, 739, 743, 751, 757, 761, 773, 787,
              809, 811, 821, 823, 827, 829, 839, 853, 857, 863, 877, 881,
              883, 907, 911, 929, 937, 941, 947, 953, 967, 977, 983, 997 ...

              This paper appears to have open access:
              http://tinyurl.com/mtvsn5
              See Section 5.1 for Cino's record holder, p = 28279,
              with 21 integral points.

              David










              [Non-text portions of this message have been removed]
            • David Broadhurst
              ... How does it compare with the brute force of issquare for the largest integer point on x^3 - y^2 = 22189 I wonder? The result is already published, but I
              Message 6 of 22 , Jun 18, 2009
              View Source
              • 0 Attachment
                --- In primenumbers@yahoogroups.com,
                cino hilliard <hillcino368@...> wrote:

                > I contacted the Pari group and got a workaround
                > for eliptic curves from Karim.

                How does it compare with the brute force of
                "issquare" for the largest integer point on
                x^3 - y^2 = 22189
                I wonder?

                The result is already published, but I thought
                that it might be an interesting test case.
                Here is an "issquare" timing:

                c=0;gettime;
                {for(x=1,10^9,if(issquare(x^3-22189),
                print(x);c=c+1;if(c==2,break)))}
                print(ceil(gettime/10^3)" seconds")

                29585
                140292677
                98 seconds

                David
              • sta staf
                ... _________________________________________________________________ Découvrez Windows Live Spaces et créez votre site Web perso en quelques clics !
                Message 7 of 22 , Jun 19, 2009
                View Source
                • 0 Attachment
                  ---
                  >
                  >
                  > the collatz function to it as in f(n) = 3n+1/2 if n is odd
                  >
                  > and f(n) = n/2 n is even.
                  >
                  > Consider the number 33
                  >
                  >
                  >
                  > 33, 100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
                  >
                  >
                  >
                  > The odd numbers are 33, 25 ;19,29 ,11 ,17 ,13 ,5 and so on.
                  >
                  >
                  >
                  > As u can see 6 numbre prime : 19 ;29; 11 ;17 ; 13; 5 .
                  >
                  >
                  >
                  > I believe that the collatz function can be used as a prime generating function though i am not very sure in this.
                  >
                  >
                  >
                  > i have found n = 111012973909 is the largest integer with all odd are prime
                  >
                  >
                  >
                  > 111012973909, 333038921728, 166519460864, 83259730432, 41629865216, 20814932608, 10407466304, 5203733152, 2601866576, 1300933288, 650466644, 325233322, 162616661, 487849984, 243924992, 121962496, 60981248, 30490624, 15245312, 7622656, 3811328, 1905664, 952832, 476416, 238208, 119104, 59552, 29776, 14888, 7444, 3722, 1861, 5584, 2792, 1396, 698, 349, 1048, 524, 262, 131, 394, 197, 592, 296, 148, 74, 37, 112, 56, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
                  >
                  >
                  >
                  > all odd are prime : 111012973909 ; 162616661 ; 1861 ; 349 ; 131 ; 197 ; 37 ; 7 ;11 ; 17 ; 13 ; ; 5
                  >
                  >
                  >
                  > rachid
                  >



                  _________________________________________________________________
                  Découvrez Windows Live Spaces et créez votre site Web perso en quelques clics !
                  http://spaces.live.com/signup.aspx

                  [Non-text portions of this message have been removed]
                • David Broadhurst
                  ... The word largest makes no sense here. The 271-digit prime (5*2^897-1)/3 manifestly contains no composite odd integer in its Collatz sequence and it would
                  Message 8 of 22 , Jun 19, 2009
                  View Source
                  • 0 Attachment
                    --- In primenumbers@yahoogroups.com,
                    sta staf <sta_staf@...> wrote:

                    > i have found n = 111012973909 is the largest integer
                    > with all odd are prime

                    The word "largest" makes no sense here.
                    The 271-digit prime (5*2^897-1)/3 manifestly contains no
                    composite odd integer in its Collatz sequence and it would be
                    easy to find larger primes by this, or a similar, construction.

                    David
                  • sta staf
                    yes but also the 271-figit prime (5*2^897-1)/3 makes no sense here you think (5*2^897-1)/3) = prime numbre !!!!!!!!!!! give me n integer 111012973909
                    Message 9 of 22 , Jun 19, 2009
                    View Source
                    • 0 Attachment
                      yes but also the 271-figit prime (5*2^897-1)/3 makes no sense here



                      you think (5*2^897-1)/3) = prime numbre !!!!!!!!!!!



                      give me n integer 111012973909 < n < 2^50 with all odd integer are prime .







                      becuse we canot test the sequnce :(5*2^897-1)/3





                      thanks

                      rachid










                      _________________________________________________________________
                      Vous voulez savoir ce que vous pouvez faire avec le nouveau Windows Live ? Lancez-vous !
                      http://www.microsoft.com/windows/windowslive/default.aspx

                      [Non-text portions of this message have been removed]
                    • David Broadhurst
                      ... to which I calmly reply, that I do so think. Here is my proof, using APR-CL in Pari-GP: N=(5*2^897-1)/3; if(isprime(N),print(OK)) OK It appears that
                      Message 10 of 22 , Jun 19, 2009
                      View Source
                      • 0 Attachment
                        --- In primenumbers@yahoogroups.com,
                        sta staf <sta_staf@...> wrote:

                        > the 271-figit prime (5*2^897-1)/3 makes no sense here
                        > you think (5*2^897-1)/3) = prime numbre !!!!!!!!!!!

                        to which I calmly reply, that I do so think.
                        Here is my proof, using APR-CL in Pari-GP:

                        N=(5*2^897-1)/3;
                        if(isprime(N),print(OK))
                        OK

                        It appears that coherence is in inverse proportion
                        to the number of exclamation marks used by the poster.

                        David (with no exclamation marks)
                      • sta staf
                        but your seqaunce :(5*2^897-1/3) this sequance of coltaze is : all Even numbers . the only odd is the ferst digit : :(5*2^897-1/3) rachid
                        Message 11 of 22 , Jun 19, 2009
                        View Source
                        • 0 Attachment
                          but your seqaunce :(5*2^897-1/3)



                          this sequance of coltaze is : all Even numbers .



                          the only odd is the ferst digit : :(5*2^897-1/3)



                          rachid










                          _________________________________________________________________
                          Découvrez toutes les possibilités de communication avec vos proches
                          http://www.microsoft.com/windows/windowslive/default.aspx

                          [Non-text portions of this message have been removed]
                        • cino hilliard
                          Hi, ... the workaround, diffcubes2(n,p) = { local(x,y,c,a); a=eval(setintersect(Set(vector(n,y,y^2)),Set(vector(n,x,x^3-p)))); a; } is useless for large n.
                          Message 12 of 22 , Jun 19, 2009
                          View Source
                          • 0 Attachment
                            Hi,


                            >>--- In primenumbers@yahoogroups.com,
                            >>cino hilliard <hillcino368@...> wrote:
                            >> I contacted the Pari group and got a workaround
                            >> for eliptic curves from Karim.

                            the workaround,

                            diffcubes2(n,p) =
                            {
                            local(x,y,c,a);
                            a=eval(setintersect(Set(vector(n,y,y^2)),Set(vector(n,x,x^3-p))));
                            a;
                            }
                            is useless for large n. Maybe someone can tweek.


                            (10:25:03) gp > diffcubes2(10000000,22189)
                            *** Set: the PARI stack overflows !
                            current stack size: 800000000 (762.939 Mbytes)
                            [hint] you can increase GP stack with allocatemem()




                            >How does it compare with the brute force of
                            >"issquare" for the largest integer point on
                            >x^3 - y^2 = 22189
                            >I wonder?


                            >c=0;gettime;
                            >{for(x=1,10^9,if(issquare(x^3-22189),
                            >print(x);c=c+1;if(c==2,break)))}
                            >print(ceil(gettime/10^3)" seconds")

                            >29585
                            >140292677
                            >98 seconds

                            elipissq(n) =
                            {
                            c=0;
                            for(x=1,10^9,if(issquare(x^3-n),
                            print(x);c=c+1;if(c==2,break)));
                            }

                            Pari 2.3.4
                            dell 8200 2.53 ghz xp pro

                            42 minutes. say what?

                            Pari 2.4.2
                            dell 8200 2.53 ghz xp pro

                            225 sec Go figure. They must have done some stuff on issquare in v2.4.2.


                            Pari 2.4.2

                            dell 6x620 3.2 ghz vista ult 64 bit op sys
                            112 sec close to David's bench.



                            Sage online calculator

                            http://www.sagenb.com/home/hillcino368/0/



                            sage: time EllipticCurve([0,0,0,0,-22189]).integral_points()


                            [(29585 : 5088706 : 1), (140292677 : 1661699554612 : 1)]
                            Time: CPU 0.22 s, Wall: 0.87 s
                            A tough and user friendly program.

                            I wonder why Pari has not implemented the Mordell algorithm. Maybe 2.4.3?
                            Perhaps someone here can write a script or show a c program with gmp that
                            does this. A gcc/gmp would be great.




                            Dénouement

                            A child playing in a room with toys could solve the riddle of the difference between
                            two squares by arranging toy blocks and counting. Later he could puzzle with numbers
                            by squaring, subtracting, noticing patterns and conjecturing. For any positive numbers
                            a,b, a^2 - b^2 = (a-b)*(a+b). Later, he finds out how to symbolically multiply
                            (a-b)*(a+b) to get a^2-b^2 proving his prior conjectures correct.


                            So a^2 - b^2 =(a-b)*(a+b) could be the third most fundamental process in arithmetic
                            the average being second and counting being first. It is that easy of a riddle.



                            Then later, the child goes into another room with more blocks. He starts making pyramids,
                            cubes, and squares. Suddenly, with a flash of insight, he about cube - square = number
                            or a^3 - b^2 = n? It is not that easy of a riddle.



                            I doubt my Mathematica I bought in 80's and Maple in the 90's does this as Sage.

                            Maybe the latest versions do.



                            [Rant]

                            It is disheartning that students can get these for $125 and seniors only get 1/2 off
                            of retail ($2000) for Mathematica.

                            [End Rant]



                            Cheers and Roebuck,

                            Cino





                            [Non-text portions of this message have been removed]
                          • David Broadhurst
                            ... As indeed I had expected :-) Sometimes brute force is best, as in my prior hack: c=0;gettime; {for(x=1,10^9,if(issquare(x^3-22189),
                            Message 13 of 22 , Jun 19, 2009
                            View Source
                            • 0 Attachment
                              --- In primenumbers@yahoogroups.com,
                              cino hilliard <hillcino368@...> wrote:

                              > the workaround,
                              > diffcubes2(n,p) =
                              > {
                              > local(x,y,c,a);
                              > a=eval(setintersect(Set(vector(n,y,y^2)),Set(vector(n,x,x^3-p))));
                              > a;
                              > }
                              > is useless for large n.

                              As indeed I had expected :-)

                              Sometimes brute force is best, as in my prior hack:

                              c=0;gettime;
                              {for(x=1,10^9,if(issquare(x^3-22189),
                              print(x);c=c+1;if(c==2,break)))}
                              print(ceil(gettime/10^3)" seconds")

                              29585
                              140292677
                              98 seconds

                              Thanks, Cino, for your fascinating thread, which
                              probes deeply into the theory of elliptic curves.

                              Best regards

                              David
                            • Peter Kosinar
                              Hello Rachid, ... It d still satisfy the property you re expecting it to satisfy: All odd terms of the sequence are primes. However, unless I m very much
                              Message 14 of 22 , Jun 19, 2009
                              View Source
                              • 0 Attachment
                                Hello Rachid,

                                Even if this was true:
                                > this sequance of coltaze is : all Even numbers .
                                > the only odd is the ferst digit : :(5*2^897-1/3)

                                It'd still satisfy the property you're expecting it to satisfy: "All odd
                                terms of the sequence are primes."

                                However, unless I'm very much mistaken, the sequence looks like this:
                                (5*2^897-1)/3, 5*2^896, 5*2^895, ..., 5*2^2, 5*2^1, 5, 8, 4, 2, 1 and as
                                far as my memory goes, the term "5" looks like an odd prime to me.

                                However, as you wanted a small example of this kind (< 2^50), consider
                                297784399189. Any objections against it?

                                Peter

                                --
                                [Name] Peter Kosinar [Quote] 2B | ~2B = exp(i*PI) [ICQ] 134813278
                              • David Broadhurst
                                ... sta staf wrote, ... Dear Rachid: Until you learn to be more disciplined, I shall no longer reply. Please isolate your error in the above
                                Message 15 of 22 , Jun 19, 2009
                                View Source
                                • 0 Attachment
                                  --- In primenumbers@yahoogroups.com,
                                  sta staf <sta_staf@...> wrote,
                                  untidily and incorrectly:

                                  > but your seqaunce :(5*2^897-1/3)
                                  > this sequance of coltaze is : all Even numbers .
                                  > the only odd is the ferst digit : :(5*2^897-1/3)
                                  > rachid

                                  Dear Rachid: Until you learn to be more disciplined,
                                  I shall no longer reply. Please isolate your error
                                  in the above statement, bearing in mind that 5
                                  is an odd prime. Then consider that you asked
                                  only for a Collatz sequence in which no odd
                                  member is composite, with which request my
                                  simple construction perfectly complies.

                                  Discipline is a stern master:
                                  I shall not allow you to "wriggle"
                                  out of the hole you have carelessly
                                  and untidily dug yourself into.

                                  You may think me stern. But please recall
                                  that mathematics is far sterner than I am.

                                  Stay well,

                                  David
                                • David Broadhurst
                                  ... After reading http://www.primepuzzles.net/puzzles/puzz_476.htm Rachid will perhaps not object to my next claim, namely that (2^1322*(5*2^897-1)/3-1)/3 is a
                                  Message 16 of 22 , Jun 20, 2009
                                  View Source
                                  • 0 Attachment
                                    --- In primenumbers@yahoogroups.com,
                                    "David Broadhurst" <d.broadhurst@...> wrote:

                                    > The 271-digit prime (5*2^897-1)/3 manifestly contains no
                                    > composite odd integer in its Collatz sequence and it would be
                                    > easy to find larger primes by this, or a similar, construction.

                                    After reading http://www.primepuzzles.net/puzzles/puzz_476.htm
                                    Rachid will perhaps not object to my next claim, namely that
                                    (2^1322*(5*2^897-1)/3-1)/3 is a 668-digit prime whose Collatz
                                    sequence contains no composite odd integer.

                                    As both Jens and I have noted, left extensibility is trivial,
                                    given enough computing power. Hence my comment to Rachid:

                                    > The word "largest" makes no sense here.

                                    David
                                  • Adriano Palma
                                    ** For Your Eyes Only ** ** High Priority ** the agent never got back to me, in spite of three different demands I assume they sold it I looked at another one
                                    Message 17 of 22 , Jun 20, 2009
                                    View Source
                                    • 0 Attachment
                                      ** For Your Eyes Only **
                                      ** High Priority **


                                      the agent never got back to me, in spite of three different demands
                                      I assume they sold it
                                      I looked at another one so I may decide to move after all.
                                      I trust & hope your vacation be goo

                                      |||||||||||||||||||||||||||||||||||||||||||||||||||||
                                      ξε ν’, γγέλλειν Λακεδαιμονίοις ἀ ὅτι τ δε
                                      κείμεθα, το ς κείνων ῥήμασι πειθόμενοι.
                                      /begin/read__>sig.file: postal address
                                      palma
                                      University of KwaZulu-Natal Philosophy
                                      3rd floor of Memorial Tower Building
                                      Howard College Campus
                                      Durban 4041
                                      South Africa
                                      Tel off: [+27] 031 2601591 (sec: Mrs. Yolanda Hordyk) [+27]
                                      031-2602292
                                      Fax [+27] 031-2603031
                                      mobile 07 62 36 23 91 calling from overseas +[27] 76 2362391
                                      EMAIL: palma@...
                                      EMAIL: palma@...
                                      MY OFFICE # IS 290@Mtb
                                      *only when in Europe*: inst. J. Nicod
                                      29 rue d'Ulm
                                      f-75005 paris france
                                      email me for details if needed at palma@...
                                      ________
                                      This e-mail message (and attachments) is confidential, and/or
                                      privileged and is intended for the
                                      use of the addressee only. If you are not the intended recipient of
                                      this e-mail you must not copy,
                                      distribute, take any action in reliance on it or disclose it to anyone.
                                      Any confidentiality or
                                      privilege is not waived or lost by reason of mistaken delivery to you.
                                      This entity is not responsible for any information not related to the
                                      business of this entity. If you
                                      received this e-mail in error please destroy the original and notify
                                      the sender.d
                                      >>> "David Broadhurst" <d.broadhurst@...> 6/20/2009 3:39 PM >>>



                                      --- In primenumbers@yahoogroups.com (
                                      mailto:primenumbers%40yahoogroups.com ),
                                      "David Broadhurst" <d.broadhurst@...> wrote:

                                      > The 271-digit prime (5*2^897-1)/3 manifestly contains no
                                      > composite odd integer in its Collatz sequence and it would be
                                      > easy to find larger primes by this, or a similar, construction.

                                      After reading http://www.primepuzzles.net/puzzles/puzz_476.htm (
                                      http://www.primepuzzles.net/puzzles/puzz_476.htm )
                                      Rachid will perhaps not object to my next claim, namely that
                                      (2^1322*(5*2^897-1)/3-1)/3 is a 668-digit prime whose Collatz
                                      sequence contains no composite odd integer.

                                      As both Jens and I have noted, left extensibility is trivial,
                                      given enough computing power. Hence my comment to Rachid:

                                      > The word "largest" makes no sense here.

                                      David



                                      Please find our Email Disclaimer here: http://www.ukzn.ac.za/disclaimer/


                                      [Non-text portions of this message have been removed]
                                    • sta staf
                                      you have the right david there is always another prime can often be extended to the left like your numbre :(2^1322*(5*2^897-1)/3-1)/3) is a Construction
                                      Message 18 of 22 , Jun 20, 2009
                                      View Source
                                      • 0 Attachment
                                        you have the right david there is always another prime can often be extended to the left
                                        like your numbre :(2^1322*(5*2^897-1)/3-1)/3)

                                        is a Construction Sequence . very easy to Making the collatz sequence contains no composite odd integer.


                                        yes ( The word "largest" makes no sense here)



                                        rachid










                                        _________________________________________________________________
                                        Vous voulez savoir ce que vous pouvez faire avec le nouveau Windows Live ? Lancez-vous !
                                        http://www.microsoft.com/windows/windowslive/default.aspx

                                        [Non-text portions of this message have been removed]
                                      • David Broadhurst
                                        ... En Angleterre, nous avons l habitude de tirer à gauche. Amitiés David
                                        Message 19 of 22 , Jun 20, 2009
                                        View Source
                                        • 0 Attachment
                                          --- In primenumbers@yahoogroups.com,
                                          sta staf <sta_staf@...> wrote:

                                          > you have the right david there is always another prime
                                          > can often be extended to the left
                                          > like your numbre : (2^1322*(5*2^897-1)/3-1)/3

                                          En Angleterre, nous avons l'habitude de tirer à gauche.

                                          Amitiés

                                          David
                                        • cino hilliard
                                          Hi, I did a brute force in gcc/gmp. http://groups.google.com/group/elliptic-curves/web/gcc-gmp-brute-force-elliptic-curve Output: n start = 22189 n stop =
                                          Message 20 of 22 , Jun 23, 2009
                                          View Source
                                          • 0 Attachment
                                            Hi,

                                            I did a brute force in gcc/gmp.

                                            http://groups.google.com/group/elliptic-curves/web/gcc-gmp-brute-force-elliptic-curve



                                            Output:

                                            n start => 22189
                                            n stop => 22189
                                            x range => 150000000
                                            max dups => 1
                                            n prime => 1
                                            29585 22189 5088706
                                            140292677 22189 1661699554612
                                            count = 2
                                            31.9994

                                            for dell 8200 2.53 ghz



                                            24.9 sec

                                            for dell gx620 3.2 ghz vista ult



                                            Working with a developer of sage I was pointed to ratpoints.

                                            >> http://www.mathe2.uni-bayreuth.de/stoll/programs/index.html, is free
                                            >> (GPL) and only requires gmp.



                                            >> ratpoints looks promising but I can't install it. Using Msys, I keep getting
                                            >> the cannot find -lgmp.

                                            I installed gmp 4.3.1 with msys which was used for the above timings.


                                            I searched long for this. There ain't much out there on it. In fact, google -lgmp
                                            gets a no hit.



                                            Still asking, how do we determine the x range for ratpoints and how does sage
                                            do it?



                                            Reply:

                                            >ratpoints takes 0.09s to fine all integral points with x<10^8! But
                                            >there is no proof, which Sage provides. Sage takes longer as the
                                            >curve has rank 5.

                                            >I cannot help with Windows problems. gmp is a standard library (Gnu
                                            >MultiPrecision) but I do not know how to use it on Windows.

                                            >Very soon ratpoints will be available within Sage -- we are just testing it now.

                                            Maybe someone in primenumbers can load ratpoints in a windows xp pro of vista
                                            platform.



                                            Anyway, the gcc/gmp brute force is considerably faster than Pari for x range 10^8-10^9.



                                            Cino





                                            To: primenumbers@yahoogroups.com
                                            From: d.broadhurst@...
                                            Date: Fri, 19 Jun 2009 23:14:46 +0000
                                            Subject: [PrimeNumbers] Re: primes of the form x^3 - y^2







                                            --- In primenumbers@yahoogroups.com,
                                            cino hilliard <hillcino368@...> wrote:

                                            > the workaround,
                                            > diffcubes2(n,p) =
                                            > {
                                            > local(x,y,c,a);
                                            > a=eval(setintersect(Set(vector(n,y,y^2)),Set(vector(n,x,x^3-p))));
                                            > a;
                                            > }
                                            > is useless for large n.

                                            As indeed I had expected :-)

                                            Sometimes brute force is best, as in my prior hack:

                                            c=0;gettime;
                                            {for(x=1,10^9,if(issquare(x^3-22189),
                                            print(x);c=c+1;if(c==2,break)))}
                                            print(ceil(gettime/10^3)" seconds")

                                            29585
                                            140292677
                                            98 seconds





                                            Recent Activity


                                            3
                                            New MembersVisit Your Group



                                            Give Back
                                            Yahoo! for Good

                                            Get inspired
                                            by a good cause.

                                            Y! Toolbar
                                            Get it Free!

                                            easy 1-click access
                                            to your groups.

                                            Yahoo! Groups
                                            Start a group

                                            in 3 easy steps.
                                            Connect with others.
                                            .








                                            [Non-text portions of this message have been removed]
                                          • David Broadhurst
                                            ... Sage is being sagacious, here. It s good that you can access such wisdom without paying big dollars for Magma. Congrats to Cino, for persistence. David
                                            Message 21 of 22 , Jun 23, 2009
                                            View Source
                                            • 0 Attachment
                                              --- In primenumbers@yahoogroups.com,
                                              cino hilliard <hillcino368@...> wrote:

                                              > Sage takes longer as the curve has rank 5.

                                              Sage is being sagacious, here.
                                              It's good that you can access
                                              such wisdom without paying big
                                              dollars for Magma.

                                              Congrats to Cino, for persistence.

                                              David
                                            Your message has been successfully submitted and would be delivered to recipients shortly.