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Re: Primality sieve challenge

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From Ben Bacarisse <ben.usenet@bsb.me.uk>
Newsgroups comp.lang.javascript
Subject Re: Primality sieve challenge
Date Tue, 19 Jan 2016 14:04:12 +0000
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Thomas 'PointedEars' Lahn <PointedEars@web.de> writes:

> Scott Sauyet wrote:
>
>> Thomas 'PointedEars' Lahn wrote:
>>> Ben Bacarisse wrote:
>>>> but all those that have the form 6k + {0, 2, 3, 4} are clearly
>>>> composite.
>>> I can see why it is so for {0, 2, 4} (all multiples of 6 are divisible by
>>> 2, so are the sums when added 2 or 4), but why also for the summand 3?
>> 
>> Because 6k + 3 is divisible by 3.
>
> Why?  I can see that it follows for k = 1 (9), k = 2 (15), k = 3 (21), and 
> for several greater k, but why for *all* k?

(6k + 3)/3 = 2k + 1 which is an integer.  If that is not enough, you'd
better say what part you doubt rather than just ask another "why?".

<snip>
>> Since all positive integers > 5 are (trivially) of one of forms `6k + 0`,
>> `6k + 1`, `6k + 2`, `6k + 3`, `6k + 4`, or `6k + 5`, and we've easily
>> demonstrated that all those of the form `6k + {0, 2, 3, 4}` are composite,
>
> Yes.
>
>> all primes must be of the form `6k + 1` or `6k + 5`.

(all suitably large primes...)

> Again, why?  If something is true for A and B, it does not follow that it is 
> not true for C ∉ {A, B}: p(A) ∧ p(B) ↛ ¬p(C); here p(X) := “X is composite 
> (not prime)”.  What relation I am missing here?

The key is that every integer > 5 is in one of the six sets Rr = { 6k +
r | k ∈ N } where r is one of 0, 1,... 5.  All the primes are there
somewhere in one or more of these sets.  But all the numbers in R0, R2,
R3 and R4 are composite.  What options are left?  There may be no
primes, of course, but if there are any, they must in either R1 or R5.

-- 
Ben.

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Thread

Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-17 01:40 -0800
  Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-17 02:20 -0800
  Re: Primality sieve challenge Ben Bacarisse <ben.usenet@bsb.me.uk> - 2016-01-17 16:24 +0000
    Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-17 09:03 -0800
      Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-17 09:17 -0800
        Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-17 09:22 -0800
          Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-17 11:32 -0800
          Re: Primality sieve challenge Stefan Weiss <krewecherl@gmail.com> - 2016-01-18 00:13 +0100
            Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-18 04:10 -0800
          Re: Primality sieve challenge Ben Bacarisse <ben.usenet@bsb.me.uk> - 2016-01-18 01:20 +0000
            Re: Primality sieve challenge "Evertjan." <exxjxw.hannivoort@inter.nl.net> - 2016-01-18 12:16 +0100
              Re: Primality sieve challenge Ben Bacarisse <ben.usenet@bsb.me.uk> - 2016-01-18 11:25 +0000
                Re: Primality sieve challenge "Evertjan." <exxjxw.hannivoort@inter.nl.net> - 2016-01-18 12:43 +0100
                Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-18 04:18 -0800
                Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-18 04:25 -0800
              Re: Primality sieve challenge Gene Wirchenko <genew@telus.net> - 2016-01-18 09:52 -0800
                Re: Primality sieve challenge "Evertjan." <exxjxw.hannivoort@inter.nl.net> - 2016-01-18 19:23 +0100
                Re: Primality sieve challenge "Michael Haufe (TNO)" <tno@thenewobjective.com> - 2016-01-18 10:48 -0800
                Re: Primality sieve challenge Gene Wirchenko <genew@telus.net> - 2016-01-18 13:09 -0800
                Re: Primality sieve challenge Scott Sauyet <scott.sauyet@gmail.com> - 2016-01-18 13:14 -0800
                Re: Primality sieve challenge "Evertjan." <exxjxw.hannivoort@inter.nl.net> - 2016-01-18 22:45 +0100
                Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-18 14:23 -0800
                Re: Primality sieve challenge Ben Bacarisse <ben.usenet@bsb.me.uk> - 2016-01-18 22:56 +0000
                Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-18 15:02 -0800
                Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-18 16:04 -0800
                Re: Primality sieve challenge Stefan Weiss <krewecherl@gmail.com> - 2016-01-19 02:42 +0100
                Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-18 23:34 -0800
                Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-19 01:34 -0800
                Re: Primality sieve challenge "Michael Haufe (TNO)" <tno@thenewobjective.com> - 2016-01-19 06:23 -0800
                Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-19 07:10 -0800
                Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-19 07:15 -0800
                Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-19 08:34 -0800
                Re: Primality sieve challenge Ben Bacarisse <ben.usenet@bsb.me.uk> - 2016-01-19 16:54 +0000
                Re: Primality sieve challenge "Michael Haufe (TNO)" <tno@thenewobjective.com> - 2016-01-19 08:57 -0800
                Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-19 10:18 -0800
                Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-19 10:59 -0800
                Re: Primality sieve challenge Stefan Weiss <krewecherl@gmail.com> - 2016-01-20 01:33 +0100
                Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-20 02:10 -0800
                Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-20 02:21 -0800
                Re: Primality sieve challenge Ben Bacarisse <ben.usenet@bsb.me.uk> - 2016-01-20 10:45 +0000
                Re: Primality sieve challenge "Michael Haufe (TNO)" <tno@thenewobjective.com> - 2016-01-20 06:43 -0800
                Re: Primality sieve challenge Scott Sauyet <scott.sauyet@gmail.com> - 2016-01-20 15:21 -0800
                Re: Primality sieve challenge Stefan Weiss <krewecherl@gmail.com> - 2016-01-21 03:18 +0100
                Re: Primality sieve challenge Jon Ribbens <jon+usenet@unequivocal.co.uk> - 2016-01-21 12:12 +0000
                Re: Primality sieve challenge Dr J R Stockton <reply1600@merlyn.demon.co.uk.invalid> - 2016-01-21 23:20 +0000
                Re: Primality sieve challenge Stefan Weiss <krewecherl@gmail.com> - 2016-01-22 19:44 +0100
                Re: Primality sieve challenge Dr J R Stockton <reply1600@merlyn.demon.co.uk.invalid> - 2016-01-23 21:50 +0000
                Re: Primality sieve challenge Stefan Weiss <krewecherl@gmail.com> - 2016-01-24 01:26 +0100
                Re: Primality sieve challenge "Michael Haufe (TNO)" <tno@thenewobjective.com> - 2016-01-23 17:54 -0800
                Re: Primality sieve challenge "Chris M. Thomasson" <nospam@no-spam.ws> - 2016-01-24 13:34 -0800
                Re: Primality sieve challenge Ben Bacarisse <ben.usenet@bsb.me.uk> - 2016-01-25 00:21 +0000
                Re: Primality sieve challenge Stefan Weiss <krewecherl@gmail.com> - 2016-01-25 03:08 +0100
                Re: Primality sieve challenge "Michael Haufe (TNO)" <tno@thenewobjective.com> - 2016-01-24 19:35 -0800
                Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-25 03:53 -0800
                Re: Primality sieve challenge Ben Bacarisse <ben.usenet@bsb.me.uk> - 2016-01-18 19:32 +0000
        Re: Primality sieve challenge Ben Bacarisse <ben.usenet@bsb.me.uk> - 2016-01-17 19:51 +0000
          Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-17 13:43 -0800
  Re: Primality sieve challenge "Chris M. Thomasson" <nospam@nospam.nospam> - 2016-01-17 12:46 -0800
    Re: Primality sieve challenge Gene Wirchenko <genew@telus.net> - 2016-01-18 09:50 -0800
      Re: Primality sieve challenge Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-01-18 20:09 +0100
        Re: Primality sieve challenge Ben Bacarisse <ben.usenet@bsb.me.uk> - 2016-01-18 19:28 +0000
          Re: Primality sieve challenge Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-01-18 21:37 +0100
            Re: Primality sieve challenge Scott Sauyet <scott.sauyet@gmail.com> - 2016-01-18 13:12 -0800
              Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-18 13:46 -0800
                Re: Primality sieve challenge jonas.thornvall@gmail.com - 2016-01-18 14:00 -0800
              Re: Primality sieve challenge Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-01-19 13:47 +0100
                Re: Primality sieve challenge Ben Bacarisse <ben.usenet@bsb.me.uk> - 2016-01-19 14:04 +0000
                Re: Primality sieve challenge Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-01-19 19:59 +0100
                Re: Primality sieve challenge Scott Sauyet <scott.sauyet@gmail.com> - 2016-01-19 10:55 -0800
                Re: Primality sieve challenge Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-01-19 20:04 +0100
                Re: Primality sieve challenge Scott Sauyet <scott.sauyet@gmail.com> - 2016-01-19 13:42 -0800
                Re: Primality sieve challenge Erwin Moller <erwinmollerusenet@xs4all.nl> - 2016-01-21 17:21 +0100
                Re: Primality sieve challenge Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-03 19:52 +0100
        Re: Primality sieve challenge Dr J R Stockton <reply1600@merlyn.demon.co.uk.invalid> - 2016-01-19 19:43 +0000

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