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Re: [PrimeNumbers] Re: Easy formula for next prime... cant make it any easier.

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  • Peter Kosinar
    Marty, ... So far, we re in agreement. ... Actually, it might be you not realizing that Lelio most likely made a typo and wrote 30 +/- 1 instead of 30n +/-
    Message 1 of 16 , Aug 1, 2010
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      Marty,

      > In 3n+2+4, the variable is n and +2+4 are the constants. In 6n+-1, again
      > n is the variable, and +- 1 are the constants.

      So far, we're in agreement.

      > Lelio may not realize it, but his 30n is actually 6n, where n is 5, so
      > declaring it as 30n is a misnnomer - shd be 30+/-1, +/-7, +/-11, +/-13
      > etc.

      Actually, it might be you not realizing that Lelio most likely made a typo
      and wrote "30 +/- 1" instead of "30n +/- 1" (and similarly for 7, 11 and
      13). That's why my question included the expressions explicitly, rather
      than referring to those from his post. So, no, it's not a misnomer.

      > True, all of these examples hit on the prime numbers, but it does fail
      > at 30+19 where 49 is not prime.

      Formally, in order to match Lelio's form, it should be written as 49 =
      2*30 - 11, rather than 1*30 + 19; but that's just an irrelevant formality.
      As far as "failing" goes, read further.

      > Secondly you misunderstand something -- all primes are 6n+/-1, not the
      > other way around that all 6n+/-1 are prime.

      Rest assured that I don't think so and I doubt that anyone in this group
      thinks so.

      > Folks out there including yourself and Lelio don't seem to understand
      > that 6n+/-1 represents ALL integers that are not multiples of 3, so of
      > course every single prime to infinity is either 6n+1, or 6n-1,

      Wrong. 6n+/-1 represents all -odd- integers not divisible by 3 and,
      consequently, it represents all primes with the exception of 2 and 3. This
      is where it differs from your form 3n+2+4, which guarantees the "not
      divisible by 3" condition, but not the "is odd" one.

      As such, the form 3n+2+4 tells us very little about the primes -- just
      that they're not divisible by 3. Yes, I believe that the ancient Greeks
      knew this, so no patentable, copyrightable or otherwiseable claim here,
      I'm afraid, unless you're going to reinvent the wheel [*].

      So, if you're trying to sieve for primes, your form allows you to cross
      out just one third of the numbers. This is actually worse than the even
      simpler form 2n+1 which tells us that primes (apart from 2, of course) are
      not even, allowing us to cross out one half of all the numbers. Shouldn't
      this latter form be seen as being even more "profound" than yours? Taken
      even further, the form "n" would be even more profound... since it cover
      -all- primes, without exceptions :-)

      The form 6n +/- 1 is interesting precisely because it's less trivial (or
      obvious, whichever you prefer) than the two mentioned in previous
      paragraph. It is also stronger than each of them -- since it combines the
      information from both. When sieving, it allows one to cross out two thirds
      of the numbers outright... which is better than the "one half" provided by
      the 2n+1 form and twice "better" than yours.

      Lelio's post just hinted that even the 6n +/- 1 form can be improved
      further. His example, 30n +/- 1, 7, 11, 13, is based on three primes, 2, 3
      and 5, guaranteeing non-divisibility by all of them. As a result, it
      allows us to cross out more than 73% of all numbers outright.

      The disadvantage of this form is that it is more complex, requiring one to
      remember four different constants (1, 7, 11, 13), rather than just one
      (1). Going further would make the situation even worse, forcing one to
      know more and more constants and the relative gain in the "quality" of the
      sieve would be lower and lower.

      All in all, the forms like 2n+1 and your 3n+2+4 (or 3n +/-1, which is the
      same) are easy to remember, but provide very little information, while
      Lelio's 30n +/- ... provide more information at the cost of more
      complexity. The 6n +/- 1 just seems like a nice compromise in between,
      making it a nice example of a not-completely-obvious "pattern" in primes.
      Other than that, there is very little "special" about it.

      My previous post was actually prompted by your statement "So you will find
      patterns at all multiples of 3n, not only at the 4 places you mention, all
      the way to infinity." which suggested that you considered Lelio's pattern
      to be invalid due to not covering all the primes; unlike your pattern,
      which covers all of them (with the exception of prime 3). In order to help
      you see that this is not so, I asked you to provide five primes which
      would be covered by your pattern, but not by Lelio's. For now, I'll assume
      that this misunderstanding was only caused by Lelio's typo -- omission of
      "n" in "30n +/- ..." and that you weren't actually suggesting that the
      expressions 30n +/- 1, +/- 7, +/- 11, +/- 13 cover all the primes apart
      from 2, 3 and 5.

      Peter

      [*] Pun not intended, but there actually -is- a wheel-relevant article in
      Wikipedia: http://en.wikipedia.org/wiki/Wheel_factorization

      > > Lelio ? Regardless how you cut it, ALL PRMES are referenced to
      > > 3n+2+4 or 6n+-1, which are actually the same two integers. So you
      > > will find patterns at all multiples of 3n, not only at the 4 places
      > > you mention, all the way to infinity. ��Marty
      >
      > Could you, please, provide five examples of primes which are of
      > the form 3n+2+4 (as you call it) and which are not of either of
      > Lelio's forms 30n +/- 1, 30n +/- 7, 30n +/- 11 and 30n +/- 13?
      >
      > Peter
      >
      > Lelio said:
      > > Next step is modulus 30
      > >
      > > Notice that all primes follow one of four patterns: (30+-1)? (30+-7)?
      > > (30+-11) and (30+-13) but I'm affraid Erathostenes
      > > (http://en.wikipedia.org/wiki/Eratosthenes) knew about it.

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    • Matteo Mattsteel Vitturi
      Peter, ... I don t thing i understand very well what you re saying about 6n+/-1 that should represent all primes. When n=20 then 6n-1=119 and 6n+1=121 and both
      Message 2 of 16 , Aug 3, 2010
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        Peter,

        > > Folks out there including yourself and Lelio don't seem to understand

        > > that 6n+/-1 represents ALL integers that are not multiples of 3, so of

        > > course every single prime to infinity is either 6n+1, or 6n-1,

        >

        > Wrong. 6n+/-1 represents all -odd- integers not divisible by 3 and,

        > consequently, it represents all primes with the exception of 2 and 3. This

        > is where it differs from your form 3n+2+4, which guarantees the "not

        > divisible by 3" condition, but not the "is odd" one.

        >



        I don't thing i understand very well what you're saying about 6n+/-1 that should represent all primes.
        When n=20 then 6n-1=119 and 6n+1=121 and both aren't primes since 121 is 11x11 and 119 is 7x17.

        Matteo.




        [Non-text portions of this message have been removed]
      • Peter Kosinar
        Matteo, ... An expression (e.g. 2n+1) represents a set S if every number from set S can be written in the form prescribed by the expression. It s not necessary
        Message 3 of 16 , Aug 3, 2010
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          Matteo,

          > > Wrong. 6n+/-1 represents all -odd- integers not divisible by 3 and,
          > > consequently, it represents all primes with the exception of 2 and 3. This
          > > is where it differs from your form 3n+2+4, which guarantees the "not
          > > divisible by 3" condition, but not the "is odd" one.
          >
          > I don't thing i understand very well what you're saying about 6n+/-1 that
          > should represent all primes.

          An expression (e.g. 2n+1) represents a set S if every number from set S
          can be written in the form prescribed by the expression. It's not
          necessary for all the numbers of that form to belong to set S.

          For example, the form 2n+1 represents all odd integers. Thus, it can also
          be used to represent each and every odd prime, odd square or odd perfect
          number -- since all of these are just subsets of the set of odd numbers.

          Peter
        • Martin Aaronson
          Peter Thanks for your comment. I admit I am wrong technically in that since 2 is the only even prime, I usually skirt around that issue by always prefacing
          Message 4 of 16 , Aug 3, 2010
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            Peter Thanks for your comment. I admit I am wrong technically in that since
            2 is the only even prime, I usually skirt around that issue by always
            prefacing my remarks by stating ³in the set of odd numbers only,² which I
            neglected to do in this case. I do that because prime 2 always obfuscates
            the issue, as it is doing in this very instance. Thanks for your interest.
            Marty



            From: Matteo Mattsteel Vitturi <mattsteel@...>
            Date: Tue, 3 Aug 2010 23:25:54 +0200
            To: <primenumbers@yahoogroups.com>
            Subject: RE: [PrimeNumbers] Re: Easy formula for next prime... cant make it
            any easier.







            Peter,

            > > Folks out there including yourself and Lelio don't seem to understand

            > > that 6n+/-1 represents ALL integers that are not multiples of 3, so of

            > > course every single prime to infinity is either 6n+1, or 6n-1,

            >

            > Wrong. 6n+/-1 represents all -odd- integers not divisible by 3 and,

            > consequently, it represents all primes with the exception of 2 and 3. This

            > is where it differs from your form 3n+2+4, which guarantees the "not

            > divisible by 3" condition, but not the "is odd" one.

            >

            I don't thing i understand very well what you're saying about 6n+/-1 that
            should represent all primes.
            When n=20 then 6n-1=119 and 6n+1=121 and both aren't primes since 121 is
            11x11 and 119 is 7x17.

            Matteo.



            [Non-text portions of this message have been removed]








            [Non-text portions of this message have been removed]
          • Martin Aaronson
            Peter Thanks for your comment. I admit I am technically wrong in that since 2 is the only even prime, I usually skirt around that special case by always
            Message 5 of 16 , Aug 3, 2010
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              Peter Thanks for your comment. I admit I am technically wrong in that
              since 2 is the only even prime, I usually skirt around that special case by
              always prefacing my remarks with ³in the set of odd numbers only,² which
              I neglected to do, much to my regret since I am getting flak from around
              the world. I do that because prime 2 always obfuscates the issue, as it is
              doing in this very instance.
              So, to correct, the locus for ALL primes except 2 is 6n+1 or 6n-1,
              which of course is not to say that all 6n+-1 are prime. Another way to
              define 6n+-1 is 3n+2+4, where both are one and the same. The distinction I
              am making is that 3n+2+4 is descriptive (to me) of all non-multiples >3 to
              infinty, while 6n+-1, makes it appear that there is something profound and
              revealing about this infinite set when in fact it has been staring us in the
              face from the very outset as 3n+2+4. If you look at in this light, it will
              all come together. Thus, the search for a pattern in 6n+1-1 is all in vain.
              A list of prime numbers (go primes.utm.edu ) has provided this information
              forever. Thanks for your interest. Regards. Marty



              From: Peter Kosinar <goober@...>
              Date: Wed, 4 Aug 2010 00:08:38 +0200 (CEST)
              To: Matteo Mattsteel Vitturi <mattsteel@...>
              Cc: <primenumbers@yahoogroups.com>
              Subject: RE: [PrimeNumbers] Re: Easy formula for next prime... cant make it
              any easier.






              Matteo,

              > > Wrong. 6n+/-1 represents all -odd- integers not divisible by 3 and,
              > > consequently, it represents all primes with the exception of 2 and 3. This
              > > is where it differs from your form 3n+2+4, which guarantees the "not
              > > divisible by 3" condition, but not the "is odd" one.
              >
              > I don't thing i understand very well what you're saying about 6n+/-1 that
              > should represent all primes.

              An expression (e.g. 2n+1) represents a set S if every number from set S
              can be written in the form prescribed by the expression. It's not
              necessary for all the numbers of that form to belong to set S.

              For example, the form 2n+1 represents all odd integers. Thus, it can also
              be used to represent each and every odd prime, odd square or odd perfect
              number -- since all of these are just subsets of the set of odd numbers.

              Peter







              [Non-text portions of this message have been removed]
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