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Re: [PrimeNumbers] Unnecessary Primes

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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 1 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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