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• ## Re: [PrimeNumbers] Unnecessary Primes

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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
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@...>
Sent: Wednesday, April 06, 2005 6:42 AM

>
>
> 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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