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Re: [PrimeNumbers] Unnecessary Primes

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  • Sudarshan Iyengar
    but you never know what is unnecessary and what is not... just that 11 doesn t alone play a role until 400 doesn t mean that it doesn t play a role at all..
    Message 1 of 3 , Apr 6, 2005
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      but you never know what is unnecessary and what is not...

      just that 11 doesn't alone play a role until 400 doesn't mean that it
      doesn't play a role at all..

      may be in does 10^10^10^10^10 number of digits down the line :-)

      -sudarshan
      ----- Original Message -----
      From: "ed pegg" <ed@...>
      To: "Sudarshan Iyengar" <sudarshan@...-edu.com>
      Sent: Wednesday, April 06, 2005 5:17 PM
      Subject: Re: [PrimeNumbers] Unnecessary Primes


      >
      > --- Sudarshan Iyengar <sudarshan@...-edu.com> wrote:
      >
      > > what is GC? what does it stand for... :-( hope you don't mind helping me
      out
      > > :-(
      >
      > http://mathworld.wolfram.com/GoldbachConjecture.html
      >
      > >
      > > -sudarshan
      > > ----- Original Message -----
      > > From: xeipon2
      > > To: primenumbers@yahoogroups.com
      > > Sent: Wednesday, April 06, 2005 10:12 AM
      > > Subject: [PrimeNumbers] Unnecessary Primes
      > >
      > >
      > >
      > > I claim that the following primes are unnecessary for GC.
      > >
      > > 11, 17, 29, 41, 59, 67, 71, 73, 89, 97, 103, 127, 137, 149, ...
      > >
      > > Here are the even numbers up to 400 expressed without these
      > > primes. Can anyone extend the list of Unnecessary Primes --
      > > or perhaps point out some even number where they are necessary?
      > > --Ed Pegg Jr
      > >
      > > {4,2,2},{6,3,3},{8,3,5},
      > > {10,3,7},{12,5,7},{14,7,7},{16,3,13},{18,5,13},
      > > {20,7,13},{22,3,19},{24,5,19},{26,3,23},{28,5,23},
      > > {30,7,23},{32,13,19},{34,3,31},{36,5,31},{38,7,31},
      > > {40,3,37},{42,5,37},{44,7,37},{46,3,43},{48,5,43},
      > > {50,3,47},{52,5,47},{54,7,47},{56,3,53},{58,5,53},
      > > {60,7,53},{62,19,43},{64,3,61},{66,5,61},{68,7,61},
      > > {70,23,47},{72,19,53},{74,13,61},{76,23,53},{78,31,47},
      > > {80,19,61},{82,3,79},{84,5,79},{86,3,83},{88,5,83},
      > > {90,7,83},{92,13,79},{94,47,47},{96,13,83},{98,19,79},
      > > {100,47,53},{102,19,83},{104,3,101},{106,5,101},{108,7,101},
      > > {110,3,107},{112,3,109},{114,5,109},{116,3,113},{118,5,113},
      > > {120,7,113},{122,13,109},{124,23,101},{126,13,113},{128,19,109},
      > > {130,23,107},{132,19,113},{134,3,131},{136,5,131},{138,7,131},
      > > {140,31,109},{142,3,139},{144,5,139},{146,7,139},{148,47,101},
      > > {150,19,131},{152,13,139},{154,3,151},{156,5,151},{158,7,151},
      > > {160,3,157},{162,5,157},{164,7,157},{166,3,163},{168,5,163},
      > > {170,3,167},{172,5,167},{174,7,167},{176,3,173},{178,5,173},
      > > {180,7,173},{182,3,179},{184,3,181},{186,5,181},{188,7,181},
      > > {190,23,167},{192,13,179},{194,3,191},{196,3,193},{198,5,193},
      > > {200,3,197},{202,3,199},{204,5,199},{206,7,199},{208,101,107},
      > > {210,13,197},{212,13,199},{214,3,211},{216,5,211},{218,7,211},
      > > {220,23,197},{222,23,199},{224,13,211},{226,3,223},{228,5,223},
      > > {230,3,227},{232,3,229},{234,5,229},{236,3,233},{238,5,233},
      > > {240,7,233},{242,3,239},{244,3,241},{246,5,241},{248,7,241},
      > > {250,23,227},{252,13,239},{254,3,251},{256,5,251},{258,7,251},
      > > {260,3,257},{262,5,257},{264,7,257},{266,3,263},{268,5,263},
      > > {270,7,263},{272,3,269},{274,3,271},{276,5,271},{278,7,271},
      > > {280,3,277},{282,5,277},{284,3,281},{286,3,283},{288,5,283},
      > > {290,7,283},{292,23,269},{294,13,281},{296,3,293},{298,5,293},
      > > {300,7,293},{302,19,283},{304,23,281},{306,13,293},{308,31,277},
      > > {310,3,307},{312,5,307},{314,3,311},{316,3,313},{318,5,313},
      > > {320,3,317},{322,5,317},{324,7,317},{326,13,313},{328,47,281},
      > > {330,13,317},{332,19,313},{334,3,331},{336,5,331},{338,7,331},
      > > {340,3,337},{342,5,337},{344,7,337},{346,53,293},{348,31,317},
      > > {350,3,347},{352,3,349},{354,5,349},{356,3,353},{358,5,353},
      > > {360,7,353},{362,3,359},{364,5,359},{366,7,359},{368,19,349},
      > > {370,3,367},{372,5,367},{374,7,367},{376,3,373},{378,5,373},
      > > {380,7,373},{382,3,379},{384,5,379},{386,3,383},{388,5,383},
      > > {390,7,383},{392,3,389},{394,5,389},{396,7,389},{398,19,379},
      > > {400,3,397}
      > >
      > >
      > >
      > >
      > >
      > > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
      > > The Prime Pages : http://www.primepages.org/
      > >
      > >
      > >
      > >
      > >
      > >
      > --------------------------------------------------------------------------
      ----
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      > >
      >
    • Jacques Tramu
      First me need some definitions : A Minimal GC-set , M-GC(N), is a set of prime numbers of minimal cardinality, which may be used to build the even numbers up
      Message 2 of 3 , Apr 6, 2005
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        First me need some definitions :
        A Minimal GC-set , M-GC(N), is a set of prime numbers of minimal
        cardinality, which may be used to
        build the even numbers up to N .They may exist several M-GC for a given N.
        See example for n = 400.

        A SET of unnecessary numbers UN-GC(N) , relatively to M-GC(N) is the
        complement of M-GC(N) relatively to the primes < N.

        For N= 400 we find the two following UN-GC(400) sets :
        { 11 17 29 41 59 67 71 73 89 97 103 137 149 151 163 179 181 191 197
        211 223 229 233 239 241 251 257 263 277 283 293 307 311 313 317 337 347 349
        353
        359 367 373 379 383 389 397 }

        { 11 17 29 41 59 67 71 73 89 97 103 127 137 149 151 163 181 191 197
        211 223 227 229 233 239 241 251 257 263 277 283 293 307 311 317 331 337 347
        353 359
        367 373 379 383 389 397 }

        Which differ after 163 ...........No reason to choose either 179 or 181
        after 163. The extension of the
        UN-GC list depends on the minimal set you choose.

        See some computations for N= 400, 1000, 10000 in annex.
        The "root" {11 17 29 41 59 67 71 73 89 97 103 127 137 149 151 163} seems
        common to all the UN-GC sets up to 10000.

        Regards,
        JT



        ----- Original Message -----
        From: "xeipon2" <ed@...>
        To: <primenumbers@yahoogroups.com>
        Sent: Wednesday, April 06, 2005 6:42 AM
        Subject: [PrimeNumbers] Unnecessary Primes


        >
        >
        > I claim that the following primes are unnecessary for GC.
        >
        > 11, 17, 29, 41, 59, 67, 71, 73, 89, 97, 103, 127, 137, 149, ...
        >
        > Here are the even numbers up to 400 expressed without these
        > primes. Can anyone extend the list of Unnecessary Primes --
        > or perhaps point out some even number where they are necessary?
        > --Ed Pegg Jr
        >

        ========================= Annex ====================
        n= 400
        card = 31 - min_gc_set _1= { 3 5 7 13 19 23 31 37 43 47 53 61 79 83 101 107
        109 113
        >>127 <<131 139 157 167 173 193 199 227 269 271 281 331 }
        UN-GC-primes_1 : { 11 17 29 41 59 67 71 73 89 97 103 137 149 151 163 179 181
        191 197
        211 223 229 233 239 241 251 257 263 277 283 293 307 311 313 317 337 347 349
        353
        359 367 373 379 383 389 397 }

        n= 400
        card = 31 - min_gc_set_2 = { 3 5 7 13 19 23 31 37 43 47 53 61 79 83 101 107
        109 113
        >>131<< 139 157 167 173 179 193 199 269 271 281 313 349 }
        UN-GC-primes_2 : { 11 17 29 41 59 67 71 73 89 97 103 127 137 149 151 163 181
        191 197
        211 223 227 229 233 239 241 251 257 263 277 283 293 307 311 317 331 337 347
        353 359
        367 373 379 383 389 397 }


        n= 1000
        card = 55 - min_gc_set = { 3 5 7 13 19 23 31 37 43 47 53 61 79 83 101 107
        109 113
        131 139 157 167 199 211 251 269 281 283 293 307 313 317 337 383 401 421 431
        439
        449 457 467 491 509 521 523 569 601 643 673 677 683 691 751 811 853 }

        UN-GC-primes : { 11 17 29 41 59 67 71 73 89 97 103 127 137 149 151 163 173
        179 181 191 193 197 223 227 229 233 239 241 257 263 271 277 311 331 347 349
        353 359 367 373 379 389 397 409 419 433 443 461 463 479 487 499 503 541 547
        557 563
        571 577 587 593 599 607 613 617 619 631 641 647 653 659 661 701 709 719 727
        733 739
        743 757 761 769 773 787 797 809 821 823 827 829 839 857 859 863 877 881 883
        887 907 911
        919 929 937 941 947 953 967 971 977 983 991 997 }


        n= 10000
        card = 223 - min_gc_set = { 3 5 7 13 19 23 31 37 43 47 53 61 79 83 101 107
        109 113 131 139 157 167
        199 211 251 269 281 283 293 307 313 337 383 401 421 431 439 449 457 491 509
        521 523 569 601 643 673
        ...
        747 8779 8821 8887 8941 9011 9029 9049 9133 9151 9311 9349 9371 9551 }

        UN-GC-primes : { 11 17 29 41 59 67 71 73 89 97 103 127 137 149 151 163 173
        179 181 191 193 197 223 227
        229 233 239 241 257 263 271 277 311 317 331 347 349 353 359 367 373 379 389
        397 409 419 433 443
        461 463 467 479 487 499 503 541 547 557 563 571 577 587 593 599 607 613 617
        619 631 641 647 653 659
        661 677 683 709 719 727 733 739 743 751 757 761 787 797 809 821 823 827 829
        853 857 859 877 887 907
        911 919 937 941 947 953 967 971 977 983 991 997 1009 1019 1021 1033 1039
        1049 1051 1061 1087 1091 1097
        .....}
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