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superpseudoprime

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  • Lio David
    The sequence number with 9 prime numbre (a,b,c,m,n,k, x,y, z)   246241 ; 262657; 279073 ;  33975937 ; 209924353 ; 4261383649 ; 487824887233 ; 138991501037953
    Message 1 of 22 , Sep 28, 2009
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      The sequence number with 9 prime numbre (a,b,c,m,n,k, x,y, z)
       
      246241 ; 262657; 279073 ;  33975937 ; 209924353 ; 4261383649 ; 487824887233 ; 138991501037953 ; 24929060818265360451708193
       
       
      a*b*c*m*n*k*x*y*z = n  is a pseudoprime  base 2  
       
      and  all composite factors of n are pesudoprime 
            we have 208 pseudoprime by 9 primes
       
       
      ab     =  246241*262657 = 64676922337        =  pseudoprime
      ac     =  68719214593                                       pseudoprime
      am    =  8366268702817                                   pseudoprime
      an     =  51691982607073                                 pseudoprime
      ak     =  1049327371113409                              pseudoprime
      ax    =   120122488057141153                          pseudoprime
      ay    =   34225406207086584673                       pseudoprime
      az    =   6138556864950480622989077152513    pseudoprime
      bc    =   73300476961                                       pseudoprime
      bm   =   8924017684609                                   pseudoprime
      bn   =   55138100785921                                  pseudoprime
      bk  =    1119282245095393                               pseudoprime
      bx  =    128130621405958081                           pseudoprime
      by  =    36507090688125621121                        pseudoprime
      bz   =     6547792327343124780164318848801                               pseudoprime
      cm  =     9481766666401                                                               pseudoprime
      cn   =    58584218964769                                                              pseudoprime
      ck   =    1189237119077377                                                           pseudoprime
      cx   =    136138754754775009                                                       pseudoprime
      cy   =     38788775169164657569                                                   pseudoprime
      cz   =     6957027789735768937339560545089                                pseudoprime
      mn  =     7132376592293761                                                          pseudoprime
      mk  =     144784502391254113                                                       pseudoprime
      mx  =     16574307635660512321                                                   pseudoprime
      my  =      4722366482800925736961                                               pseudoprime
      mz  =      846988199830552335989529107751841                            pseudoprime
       nk  =      894568205401104097                                                      pseudoprime
      nx  =      102406323829685485249                                                  pseudoprime
      ny  =      29177700927891111969409                                               pseudoprime
      nz  =      5233216963172006375136630160324129                           pseudoprime
      kx  =      2078808998029975053217                                                pseudoprime
      ky  =      592296109873099442630497                                             pseudoprime
      kz  =      106232292155882567572000547769536257                        pseudoprime
      xy  =      67803513320184824678154049                                         pseudoprime
      xz  =      12161016282494898169024647192447199969                    pseudoprime
      yz  =      3464927582597123310724584984291374048929                 pseudoprime
      abcm   =      613251486300372864299137                                              pseudoprime
      abcn    =      3789046980158167152145153                                            pseudoprime
      abck    =      76916196790845133322670049                                           pseudoprime
      abcx    =      8805035668330472737001476033                                       pseudoprime
      abcy    =      2508738599163416594755412218753                                  pseudoprime
      abcz    =     449959146053091093311287776772430639752993                p
      abmn   =     461300166938020292886639457                                         pseudoprime
      abmk  =       9364216016760333054532822081                                       pseudoprime
      abmx =       1071975207741161047082948614177                                  pseudoprime
      abmy  =       305428130254967320121131087397857                               pseudoprime
      abmz =       54780590020796069994608704147266725025772417            pseudoprime
      abnk  =       57857918345876673615721514689                                      pseudoprime
      abnx  =      6623325853150240544585732106913                                   pseudoprime
      abny   =       1887123896885426286038037012788833                             pseudoprime
      abnz   =       338468367099746845497180716643175550080169473          pseudoprime
      abkx   =       134450968119041482514963351008129                               pseudoprime
      abky   =       38307889498769671576490641456711489                            pseudoprime
      abkz =         6870777709447511120546434083811755068494672609         pseudoprime
      abxy =   pseudoprime
      abxz =   pseudoprime
      abyz=    pseudoprime
      acmn = pseudoprime
      acmk = pseudoprime
      acmx = pseudoprime
      acmy=  pseudoprime
      acmz=  pseudoprime
      acnk =  pseudoprime
      acnx=   pseudoprime
      acny =  pseudoprime
      acnz =  pseudoprime
      ackx  = pseudoprime
      acky  = pseudoprime
      ackz  = pseudoprime
      acxy  = pseudoprime
      acxz =  pseudoprime
      acyz  = pseudoprime
      amnk =    pseudoprime
      amnx =    pseudoprime
      amny  =  pseudoprime
      amnz =   pseudoprime
      amkx  =  pseudoprime
      amky  =  pseudoprime
      amkz  =  pseudoprime
      amxy  =  pseudoprime
      amxz  =  pseudoprime
      amyz =   pseudoprime
      ankx  =  pseudoprime
      anky  =  pseudoprime
       ankz =  pseudoprime
       anxy  =  pseudoprime
      anxz  =   pseudoprime
      anyz   =  pseudoprime
      akxy  =   pseudoprime
      akyz  =   pseudoprime
      axyz =    pseudoprime
      abmnk  =   pseudoprime
      abmkx =    pseudoprime
      abmxy =    pseudoprime
      abmyz  =   pseudoprime
      abnkx  =    pseudoprime
      abnxy  =    pseudoprime
      abnyz  =    pseudoprime
      abkxy  =    pseudoprime
      abkyz  =    pseudoprime
      abxyz  =    pseudoprime
      abcmn  =   pseudoprime
      abcmk =      pseudoprime
      abcmx =     pseudoprime
      abcmy  =    pseudoprime
      abcmz   =   pseudoprime
      acmnk  =    pseudoprime
      acmnx =    pseudoprime
      acmny  =   pseudoprime
      acmnz  =    pseudoprime
      amnkx =    pseudoprime
      amnky  =   pseudoprime
      amnkz  =    pseudoprime
      ankxy  =     pseudoprime
      ankxz  =     pseudoprime
      ankyz =      pseudoprime
      akxyz  =     pseudoprime
      abcmnk =     pseudoprime
      abcmnx =    pseudoprime
      abcmny  =   pseudoprime
      abcmnz  =   pseudoprime
      abcmkx  =   pseudoprime
      abcmky  =   pseudoprime
      abcmkz =    pseudoprime
      abcmxy  =   pseudoprime
      abcmyz  =   pseudoprime
      abcnxy  =    pseudoprime
      abcnyz =     pseudoprime
      abckxy  =    pseudoprime
      abckyz  =    pseudoprime
      abcxyz =     pseudoprime
      abmnkx  =  pseudoprime
      abmnky  =  pseudoprime
      abmnkz  =  pseudoprime
      abnkxy  =   pseudoprime
      abnkxz  =   pseudoprime
      abkxyz =      pseudoprime
      abcmnkx =   pseudoprime
      abcmnky =   pseudoprime
      abcmnkz =   pseudoprime
      abcmnxy  =  pseudoprime
      abcmny   =   pseudoprime
      cmnkxyz =   pseudoprime
      bmnkxyz =   pseudoprime
      bcnkxyz =    pseudoprime
      bcmkxyz =   pseudoprime
      bcmnxyz =   pseudoprime
      bcmnkyz =   pseudoprime
      bcmnkxz =   pseudoprime
      amnkxyz =   pseudoprime
      acnkxyz  =   pseudoprime
      acmkxyz =     pseudoprime
      acmnxyz  =   pseudoprime
      acmnkyz =    pseudoprime
      acmnkxz =    pseudoprime
      acmnkxy =    pseudoprime
      abnkxyz  =  pseudoprime
      abmkxyz =  pseudoprime
      abmnxyz =  pseudoprime
      abmnkyz =  pseudoprime
      abmnkxz =  pseudoprime
      abmnkxy =  pseudoprime
      abckxyz =  pseudoprime
      abcnxyz =  pseudoprime
      abcnkyz =  pseudoprime
      abcnkxz =  pseudoprime
      abcnkxy =  pseudoprime
      abcmxyz =  pseudoprime
      abcmkyz =  pseudoprime
      abcmkxz =  pseudoprime
      abcmkxy =  pseudoprime
      abcmnyz =  pseudoprime
      abcmnxz =  pseudoprime
      abcmnxy =  pseudoprime
      abcmnkz =  pseudoprime
      abcmnky =  pseudoprime
      abcmnkx =  pseudoprime
      abcmnkxy =  pseudoprime
      abcmnkxz =  pseudoprime
      abcmnkyz =  pseudoprime
      abcmnxyz =  pseudoprime
      abcmkxyz =  pseudoprime
      abcnkxyz  =  pseudoprime
      abmnkxyz =  pseudoprime
      acmnkxyz =  pseudoprime
      bcmnkxyz =  pseudoprime
       
        a*b*c*m*n*k*x*y*z = 927278484441774426341627658843620260247978492902026521810901981921188130379449817967064027073 is a pesudoprime base 2
        
      conclusion :  n is a psudoprime with 9 prime factor    
       all  composite  factor are pseudoprime.

      For more information visit   http://groups.google.co.ma/group/-sequence-of-prime-numbers-/web/superpseudpprime?hl=fr
      pour plus d’informations consultez la page 
      thanks




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    • Lio David
      sory for all if the message is not in order you can Visithttp://groups.google.co.ma/group/-sequence-of-prime-numbers-/web/superpseudpprime?hl=fr   thanks
      Message 2 of 22 , Sep 29, 2009
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        sory for all if the message is not in order
        you can Visithttp://groups.google.co.ma/group/-sequence-of-prime-numbers-/web/superpseudpprime?hl=fr
         
        thanks




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      • djbroadhurst
        ... 2^9 - 10 = 502 products are base-2 pseudoprimes: {a=[246241,262657,279073,33975937,209924353,4261383649,
        Message 3 of 22 , Sep 29, 2009
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          --- In primenumbers@yahoogroups.com,
          Lio David <maths_forall@...> wrote:

          > sory for all if the message is not in order

          \\ 2^9 - 10 = 502 products are base-2 pseudoprimes:

          {a=[246241,262657,279073,33975937,209924353,4261383649,
          487824887233,138991501037953,24929060818265360451708193];
          c=0;for(k=1,511,b=binary(512+k);p=prod(k=1,9,a[k]^b[k+1]);
          if(!isprime(p)&&Mod(2,p)^p==2,c+=1));print(c)}

          502

          David
        • maximilian_hasler
          ... nice find - congrats to the original poster ! ... Your duplicate usage of the loop variable k is original but maybe a little confusing... note that
          Message 4 of 22 , Sep 29, 2009
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            On Tue, Sep 29, 2009 at 6:20 AM, djbroadhurst wrote:
            > \\ 2^9 - 10 = 502 products are base-2 pseudoprimes:

            nice find - congrats to the original poster !

            > {a=[246241,262657,279073,33975937,209924353,4261383649,
            > 487824887233,138991501037953,24929060818265360451708193];
            > c=0;for(k=1,511,b=binary(512+k);p=prod(k=1,9,a[k]^b[k+1]);
            > if(!isprime(p)&&Mod(2,p)^p==2,c+=1));print(c)}

            Your duplicate usage of the loop variable k is original but maybe a little confusing... note that bittest() can replace the "binary"-hack:

            c=0;for(k=1,511,p=prod(j=1,9,a[j]^bittest(k,j-1));if(!isprime(p)&&Mod(2,p)^p==2,c+=1));print(c)

            or simply:

            sum(k=1,2^#a-1, !isprime(p=prod(j=1,#a,a[j]^bittest(k,j-1))) && Mod(2,p)^p==2)

            PS: are all of these primes in
            http://www.research.att.com/~njas/sequences/A104885
            Primes whose logarithms are known to possess binary BBP formulas.

            and if so, is there a simple explanation ?

            Maximilian
          • djbroadhurst
            ... Interesting question! No idea, my end. The most relevant OEIS sequence is http://www.research.att.com/~njas/sequences/A050217 leading to
            Message 5 of 22 , Sep 29, 2009
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              --- In primenumbers@yahoogroups.com, "maximilian_hasler" <maximilian.hasler@...> wrote:

              > PS: are all of these primes in
              > http://www.research.att.com/~njas/sequences/A104885
              > Primes whose logarithms are known to possess binary BBP formulas.

              Interesting question! No idea, my end.

              The most relevant OEIS sequence is
              http://www.research.att.com/~njas/sequences/A050217
              leading to
              http://en.wikipedia.org/wiki/Super-Poulet_number
              which makes it clear that Cunningham project
              factors are useful here.

              http://www.leyland.vispa.com/numth/factorization/cunningham/2+.txt
              > 108 33975937.138991501037953
              > 216 209924353.4261383649.24929060818265360451708193

              Here are the indices of that set of 9:

              {a=[246241,262657,279073,33975937,209924353,4261383649,
              487824887233,138991501037953,24929060818265360451708193];
              for(k=1,9,p=a[k];for(j=1,1000,
              if(Mod(2,p)^j==1,print([k,j]);break)))}

              [1, 108]
              [2, 27]
              [3, 108]
              [4, 216]
              [5, 432]
              [6, 432]
              [7, 144]
              [8, 216]
              [9, 432]

              So the "Super Poulet" number is simply
              a divisor of 2^432-1, in this case.

              David
            • maximilian_hasler
              ... thanks for pointing out these. ... I also remarked this - by a simple google search for the larger numbers.... Just for information, I consider submitting
              Message 6 of 22 , Sep 29, 2009
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                --- In primenumbers@yahoogroups.com, "djbroadhurst" wrote:

                > http://www.research.att.com/~njas/sequences/A050217
                > http://en.wikipedia.org/wiki/Super-Poulet_number

                thanks for pointing out these.

                > which makes it clear that Cunningham project
                > factors are useful here.

                I also remarked this - by a simple google search for the larger numbers....

                Just for information, I consider submitting

                294409, 1398101, 1549411, 1840357,12599233, 13421773, 15162941, 15732721, 28717483, 29593159, 61377109, 66384121, 67763803, 74658629, 78526729, 90341197, 96916279, 109322501, 135945853, 153369061, 157010389, 163442551, 206453509, 221415781, 231927781, 271682651, 351593899, 367632301, 434042801, 457457617, 464955857, 491738801, 516045197, 536870911, 604611019, 611097401, 612006253, 630622753, 762278161

                to the OEIS, as
                A165777 Super-pseudoprimes (to base 2)
                COMMENT:
                A super-pseudoprime (to base b) is a pseudoprime to base b whose composite divisors are again pseudoprimes to base b, and which is not a semiprime (so that it has at least one composite proper divisor).


                All super-pseudoprimes below 10^9 are listed above, they all have omega(n)=3. (Actually I must verify that there are none with omega=2 and bigomega>2 - by error my script only considered those with omega>2 instead of bigomega>2.)

                What is the least super-pseudoprime with bigomega > 3 ?

                Maximilian
              • djbroadhurst
                ... The OP pointed to a claim that le plus grand nombre superpseudoprime connu est 92727848.....64027073 avec 9 factors premier. The plus grand ... connu
                Message 7 of 22 , Sep 29, 2009
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                  "maximilian_hasler" <maximilian.hasler@...> wrote:

                  > nice find - congrats to the original poster !

                  The OP pointed to a claim that
                  "le plus grand nombre superpseudoprime connu est
                  92727848.....64027073 avec 9 factors premier."

                  The "plus grand ... connu" claim is rather weird:
                  to make a bigger one, we simply multiply the
                  original number by 433.

                  The construction is very simple, given the
                  Cunningham project. We multiply the listed prime
                  factors of 2^432 - 1 that are congruent
                  to 1 mod 432. Then we have a so-called "Super Poulet"
                  number, N, with 10 (not 9) distinct prime divisors.

                  Proof: By construction, every divisor d|N satisfies
                  d = 1 mod 432 and 2^432 = 1 mod d.
                  Hence 2^d = 2 mod d and we are done.

                  More like Cunningham fried chicken than Super Poulet?

                  David
                • djbroadhurst
                  ... Here s some more fast food from the good Colonel: 70171342151*Phi(410,2)*Phi(1025,4) is a 541-digit Super Poulet number with 2^16 - 17 = 65519 pseudoprime
                  Message 8 of 22 , Sep 29, 2009
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                    --- In primenumbers@yahoogroups.com,
                    "djbroadhurst" <d.broadhurst@...> wrote:

                    > More like Cunningham fried chicken than Super Poulet?

                    Here's some more fast food from the good Colonel:

                    70171342151*Phi(410,2)*Phi(1025,4)
                    is a 541-digit Super Poulet number with
                    2^16 - 17 = 65519 pseudoprime divisors

                    Incidentally, Cunningham really was a colonel. See
                    http://www.nature.com/nature/journal/v89/n2213/abs/089086c0.html

                    David
                  • djbroadhurst
                    ... The Wieferich constructions 1093^2 * 4733 3511^2 * 1969111 are Super Poulet numbers that are not square-free. David
                    Message 9 of 22 , Sep 29, 2009
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                      --- In primenumbers@yahoogroups.com,
                      "maximilian_hasler" <maximilian.hasler@...> wrote:

                      > I must verify that there are none with
                      > omega=2 and bigomega>2

                      The Wieferich constructions
                      1093^2 * 4733
                      3511^2 * 1969111
                      are Super Poulet numbers that are not square-free.

                      David
                    • djbroadhurst
                      ... (4^21-1)/21^2 = 9972894583 == 43*127*337*5419 with certainty. The least with bigomega 4 may be Phi(51,4)/11119 = 1264022137981459 =
                      Message 10 of 22 , Sep 29, 2009
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                        --- In primenumbers@yahoogroups.com,
                        "maximilian_hasler" <maximilian.hasler@...> asked:

                        > What is the least super-pseudoprime with bigomega > 3 ?

                        (4^21-1)/21^2 = 9972894583 == 43*127*337*5419
                        with certainty.

                        The least with bigomega > 4 may be
                        Phi(51,4)/11119 = 1264022137981459 = 103*307*2143*2857*6529
                        but I am not sure of that.

                        David
                      • djbroadhurst
                        ... The largest known Super Poulet is quite large: (4^(2^43112609 - 1) - 1)/3 David
                        Message 11 of 22 , Sep 30, 2009
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                          --- In primenumbers@yahoogroups.com,
                          Lio David <maths_forall@...> wrote:

                          > There's Not Sense for the search

                          The largest known Super Poulet is quite large:

                          (4^(2^43112609 - 1) - 1)/3

                          David
                        • Lio David
                          yes the superpoulet is the cousin of  the superpseudoprime lol a simple way to make the superpoulet n is even numbre   factorize 2^n-1  and select the
                          Message 12 of 22 , Sep 30, 2009
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                            yes the superpoulet is the cousin of  the superpseudoprime lol

                            a simple way to make the superpoulet
                            n is even numbre

                              factorize 2^n-1  and select the factor witch k = 1 mod n


                             example n = 100
                              2^100-1 = 5*5*5*11*31*41*101*251*601*1801*4051*8101*268501

                              so the test 101*601      is superpoulet
                                              101*1801    is superpoulet
                                              601*1801    is superpoulet
                                              


                            rachid




                            [Non-text portions of this message have been removed]
                          • djbroadhurst
                            ... Bien sûr. Mais vous êtes venu ici trop tard: nous avons déjà décidé tout cela, avec des exemples énormes. Amitiés David
                            Message 13 of 22 , Sep 30, 2009
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                              --- In primenumbers@yahoogroups.com,
                              Lio David <maths_forall@...> wrote:

                              > a simple way to make the superpoulet
                              > n is even numbre
                              > factorize 2^n-1 and select the factors with
                              > k = 1 mod n

                              Bien sûr. Mais vous êtes venu ici trop tard:
                              nous avons déjà décidé tout cela, avec des
                              exemples énormes.

                              Amitiés

                              David
                            • j_chrtn
                              ... Bonsoir à tous, bonsoir David, Cela fait plaisir de pouvoir lire quelques mots dans la langue de Molière sur ce forum où l on discute majoritairement
                              Message 14 of 22 , Sep 30, 2009
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                                --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
                                >
                                > --- In primenumbers@yahoogroups.com,
                                > Lio David <maths_forall@> wrote:
                                >
                                > > a simple way to make the superpoulet
                                > > n is even numbre
                                > > factorize 2^n-1 and select the factors with
                                > > k = 1 mod n
                                >
                                > Bien sûr. Mais vous êtes venu ici trop tard:
                                > nous avons déjà décidé tout cela, avec des
                                > exemples énormes.
                                >
                                > Amitiés
                                >
                                > David
                                >

                                Bonsoir à tous, bonsoir David,

                                Cela fait plaisir de pouvoir lire quelques mots dans la langue de Molière sur ce forum où l'on discute majoritairement dans celle de Shakespeare.

                                On y parle de poulets (et autres volailles ?) avec des super pouvoirs. Espérons juste que ces drôles de gallinacés ne soient pas porteurs du virus H5N1 !

                                ;-)

                                Jean-Louis
                              • Yann Guidon
                                ... bah moi ce qui m importe surtout c est le SNR du groupe (auf english: What matters for me is the Signal-to-Noise ratio of the group more thant the
                                Message 15 of 22 , Sep 30, 2009
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                                  j_chrtn wrote:
                                  > --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
                                  >> Bien sûr. Mais vous êtes venu ici trop tard:
                                  >> nous avons déjà décidé tout cela, avec des
                                  >> exemples énormes.
                                  >>
                                  >> Amitiés
                                  >>
                                  >> David
                                  >
                                  > Bonsoir à tous, bonsoir David,
                                  >
                                  > Cela fait plaisir de pouvoir lire quelques mots dans la langue de Molière sur ce forum où l'on discute majoritairement dans celle de Shakespeare.

                                  bah moi ce qui m'importe surtout c'est le SNR du groupe

                                  (auf english: What matters for me is the Signal-to-Noise ratio of the group
                                  more thant the language)

                                  > Jean-Louis
                                  yg
                                • Yann Guidon
                                  ... s/under/oder/ ? grüsse, ... yg
                                  Message 16 of 22 , Sep 30, 2009
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                                    Norman Luhn wrote:
                                    > Na was denn nun? Englisch, Franzoesich under Deutsch ?
                                    s/under/oder/ ?

                                    grüsse,

                                    > Hehe
                                    yg
                                  • Norman Luhn
                                    Na was denn nun? Englisch, Franzoesich under Deutsch ? Hehe ... Von: Yann Guidon Betreff: Re: [PrimeNumbers] Re: superpseudoprime An:
                                    Message 17 of 22 , Sep 30, 2009
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                                      Na was denn nun? Englisch, Franzoesich under Deutsch ?

                                      Hehe

                                      --- Yann Guidon <whygee@...> schrieb am Mi, 30.9.2009:

                                      Von: Yann Guidon <whygee@...>
                                      Betreff: Re: [PrimeNumbers] Re: superpseudoprime
                                      An: primenumbers@yahoogroups.com
                                      Datum: Mittwoch, 30. September 2009, 22:26






                                       





                                      j_chrtn wrote:

                                      > --- In primenumbers@ yahoogroups. com, "djbroadhurst" <d.broadhurst@ ...> wrote:

                                      >> Bien sûr. Mais vous êtes venu ici trop tard:

                                      >> nous avons déjà décidé tout cela, avec des

                                      >> exemples énormes.

                                      >>

                                      >> Amitiés

                                      >>

                                      >> David

                                      >

                                      > Bonsoir à tous, bonsoir David,

                                      >

                                      > Cela fait plaisir de pouvoir lire quelques mots dans la langue de Molière sur ce forum où l'on discute majoritairement dans celle de Shakespeare.



                                      bah moi ce qui m'importe surtout c'est le SNR du groupe



                                      (auf english: What matters for me is the Signal-to-Noise ratio of the group

                                      more thant the language)



                                      > Jean-Louis

                                      yg































                                      [Non-text portions of this message have been removed]
                                    • djbroadhurst
                                      ... Pour être tout à fait complet, il me faut ajouter que le plus grand Poulet connu : N = (4^(2^43112609 - 1) - 1)/3 est vraiment un super-Poulet, parce
                                      Message 18 of 22 , Sep 30, 2009
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                                        --- In primenumbers@yahoogroups.com,
                                        "j_chrtn" <j_chrtn@...> wrote:

                                        > Cela fait plaisir de pouvoir lire quelques mots dans
                                        > la langue de Molière sur ce forum où l'on discute
                                        > majoritairement dans celle de Shakespeare.

                                        > On y parle de poulets (et autres volailles ?) avec des
                                        > super pouvoirs.

                                        Pour être tout à fait complet, il me faut ajouter que
                                        "le plus grand Poulet connu":
                                        N = (4^(2^43112609 - 1) - 1)/3
                                        est vraiment un super-Poulet, parce que chaque diviseur
                                        d|N possède la propriété que 2^d = 2 mod d,
                                        et il y'en a, au moins, 4 tels diviseurs.
                                        Mais on ne sait point si N est un super-pseudopremier,
                                        parce que il nous reste les deux
                                        possibilités (tout à fait éloignées) que
                                        2^(2^43112609 - 1) - 1
                                        est un permier de Mersenne et que
                                        (2^(2^43112609 - 1) + 1)/3
                                        est un premier de Wagstaff.

                                        Comment pouvons nous exclure au moins
                                        une de ces possibilités bizarres?

                                        Il me semble extrêmement difficile, à ce moment,
                                        parce que chaque premier p|N est nécessairement
                                        plus grand que le plus grand premier connu.

                                        "Plus l'obstacle était grand, plus fort fut le désir."

                                        David (en l'esprit d'une entente cordiale)
                                      • djbroadhurst
                                        ... Ich meine etwas ganz einfach: kannst Du zeigen dass 2^(2^43112609 - 1) - 1 ist keine Primzahl, oder dass (2^(2^43112609 - 1) + 1)/3 ist keine Primzhal?
                                        Message 19 of 22 , Sep 30, 2009
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                                          --- In primenumbers@yahoogroups.com,
                                          Norman Luhn <nluhn@...> wrote:

                                          > Na was denn nun? Englisch, Franzoesich under Deutsch ?

                                          Ich meine etwas ganz einfach: kannst Du zeigen dass
                                          2^(2^43112609 - 1) - 1
                                          ist keine Primzahl, oder dass
                                          (2^(2^43112609 - 1) + 1)/3
                                          ist keine Primzhal?

                                          Schwierig, nicht wahr?

                                          Alles Gute!

                                          David
                                        • djbroadhurst
                                          ... Oh dear, I am out of my linguistic depth: I forgot that Norman speaks backwards :-) ... Translation: Can you show that 2^(2^43112609 - 1) - 1 is not prime,
                                          Message 20 of 22 , Sep 30, 2009
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                                            --- In primenumbers@yahoogroups.com,
                                            "djbroadhurst" <d.broadhurst@...> wrote:

                                            > kannst Du zeigen dass
                                            > 2^(2^43112609 - 1) - 1
                                            > ist keine Primzahl, oder dass
                                            > (2^(2^43112609 - 1) + 1)/3
                                            > ist keine Primzahl?

                                            Oh dear, I am out of my linguistic depth:
                                            I forgot that Norman speaks backwards :-)

                                            Correction:

                                            > kannst Du zeigen dass
                                            > 2^(2^43112609 - 1) - 1
                                            > keine Primzahl ist, oder dass
                                            > (2^(2^43112609 - 1) + 1)/3
                                            > keine Primzahl ist?

                                            Translation:

                                            Can you show that
                                            2^(2^43112609 - 1) - 1
                                            is not prime, or that
                                            (2^(2^43112609 - 1) + 1)/3
                                            is not prime?

                                            If (like me) you cannot, then you also
                                            cannot show that the proven Super Poulet
                                            4^(2^43112609 - 1) - 1)/3
                                            is also a super-pseudoprime,
                                            according to the strict definition of Maximilian.

                                            David (relieved to return to English)
                                          • djbroadhurst
                                            86225219*5259738299*5949540043*12482997260297*(2^43112609-1) is the largest known completely factorized superpseudoprime, discovered by Edson Smith
                                            Message 21 of 22 , Oct 1, 2009
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                                              86225219*5259738299*5949540043*12482997260297*(2^43112609-1)
                                              is the largest known completely factorized superpseudoprime,
                                              discovered by Edson Smith
                                              http://primes.utm.edu/bios/page.php?id=1498
                                              and Alex Kruppa
                                              http://www.mersenneforum.org/showpost.php?p=142690&postcount=712

                                              Puzzle: Find another superpseudoprime with at least
                                              a million decimal digits and precisely 32 divisors.

                                              Hint: This may be done by judicious googling.

                                              David Broadhurst
                                            • djbroadhurst
                                              A base-b superpseudoprime is a non-semiprime composite number all of whose composite divisors are base-b pseudoprimes.
                                              Message 22 of 22 , Oct 2, 2009
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                                                A base-b superpseudoprime is a non-semiprime composite
                                                number all of whose composite divisors are base-b pseudoprimes.

                                                1340753*2011129*803278043*(89^11971-1)/88 is a gigantic
                                                base-89 superpseudoprime with precisely 11 composite divisors.

                                                Puzzle 89: For a base with 89 > b > 2, find a gigantic
                                                base-b superpseudoprime with precisely 26 composite divisors.

                                                Hint: For the meat, see http://aruljohn.com/Bible/kjv/luke/12/42

                                                David Broadhurst
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