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Re: Set of prime numbers

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  • maximilian_hasler
    ... Of course, A.A contains 1.A = { 1,..., N } and B.B contains all composite numbers up to N, and { primes } = { all numbers } { 1, composite numbers }. But
    Message 1 of 10 , Dec 1, 2009
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      --- In primenumbers@yahoogroups.com, "murat.cagliyan" :

      > A = (1,2,3,4, ... N) and B = (2,3,4, ... N) get. Number N of
      > up to a set of primes P,
      > P = (A x A) / (B x B) / (1) shows.
      > The number of primes up to here sum of N also can be found easily.

      Of course, A.A contains 1.A = { 1,..., N }
      and B.B contains all composite numbers up to N, and

      { primes } = { all numbers } \ { 1, composite numbers }.

      But your method has already been published by Erastothenes some 2200 years ago, and was certainly known some 1000 (maybe 25 000) years earlier.

      Maximilian
    • djbroadhurst
      ... I m intrigued by the certainly ; I would have said probably for 1k BCE. Do you have a source for this reasonable supposition? I m rather skeptical about
      Message 2 of 10 , Dec 1, 2009
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        --- In primenumbers@yahoogroups.com,
        "maximilian_hasler" <maximilian.hasler@...> wrote:

        > published by Eratosthenes some 2200 years ago,
        > and was certainly known some 1000
        > (maybe 25 000) years earlier.

        I'm intrigued by the "certainly";
        I would have said "probably" for 1k BCE.
        Do you have a source for this reasonable supposition?

        I'm rather skeptical about the "maybe";
        how can we know anything about maths circa 25k BCE ?
        As far as I know, the earliest proto-mathematical bone
        artefacts that we have (found in Ishango) originate
        after 10k BCE. How do you get back to 25k BCE?

        David
      • Phil Carmody
        ... I d have said definitely for 3k BCE. Base 60 just screams knowledge of divisibility properties. ... Ditto. ... Well, not everything that is known is
        Message 3 of 10 , Dec 1, 2009
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          -- On Tue, 12/1/09, djbroadhurst <d.broadhurst@...> wrote:
          > --- In primenumbers@yahoogroups.com,
          > "maximilian_hasler" <maximilian.hasler@...> wrote:
          > > published by Eratosthenes some 2200 years ago,
          > > and was certainly known some 1000
          > > (maybe 25 000) years earlier.
          >
          > I'm intrigued by the "certainly";
          > I would have said "probably" for 1k BCE.

          I'd have said "definitely" for >3k BCE. Base 60 just screams
          knowledge of divisibility properties.

          > Do you have a source for this reasonable supposition?
          >
          > I'm rather skeptical about the "maybe";

          Ditto.

          > how can we know anything about maths circa 25k BCE ?
          > As far as I know, the earliest proto-mathematical bone
          > artefacts that we have (found in Ishango) originate
          > after 10k BCE. How do you get back to 25k BCE?

          Well, not everything that is known is documented in a lastable form. Two traders not wanting to exchange the entirety of their wares might need to be able to halve, and might have problems halving 5 goats or 3 bushels. Does that mean they have an understanding of divisibility by two? In an abstract sense, quite probably not, even if they're aware that some numbers don't have that property.

          The Ishango bone with its supposed patterns of prime (and composite) numbers gives no clear indication of knowledge of any divisibility (or indivisibility) properties.

          Phil, or not Phil
        • maximilian_hasler
          ... I had similar ideas as Phil on this ... ... It seems that the first estimation of about 9k BC has been revised to ~ 25k (Wikipedia says 20k, elsewhere I
          Message 4 of 10 , Dec 1, 2009
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            --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
            >
            >
            >
            > --- In primenumbers@yahoogroups.com,
            > "maximilian_hasler" <maximilian.hasler@> wrote:
            >
            > > published by Eratosthenes some 2200 years ago,
            > > and was certainly known some 1000
            > > (maybe 25 000) years earlier.
            >
            > I'm intrigued by the "certainly";
            > I would have said "probably" for 1k BCE.
            > Do you have a source for this reasonable supposition?

            I had similar ideas as Phil on this ...

            > I'm rather skeptical about the "maybe";
            > how can we know anything about maths circa 25k BCE ?
            > As far as I know, the earliest proto-mathematical bone
            > artefacts that we have (found in Ishango) originate
            > after 10k BCE. How do you get back to 25k BCE?

            It seems that the first estimation of about 9k BC has been revised to ~ 25k (Wikipedia says 20k, elsewhere I found 25k)
            but this is still 10k less than the Lebombo bone, -- although both (esp. the latter) may be more of a calendar than a table of primes... (even if they have "29" in base 1 written on it...)

            But if someone writes (or carves) a preprint about the number of days of a lunar cycle, then it should take less than 10 000 yrs to him
            (or her) to publish a paper (or bone) about composite numbers
            (i.e.: products, as the O.P. observed) and their complement...
            (or at least "know" about it, which was all I "claimed"...)

            Maximilian
          • murat.cagliyan
            Thank you, Maximilian; but I told him a little differently. I mean, in fact, become getrimekti formula. Erastothenes, as this method performed. I brought it
            Message 5 of 10 , Dec 1, 2009
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              Thank you, Maximilian;
              but I told him a little differently. I mean, in fact, become getrimekti formula. Erastothenes, as this method performed. I brought it into the formula and I thought I could use proof of Goldbach's conjecture. also of prime test, an easier method, moreover contain certainty.
              While investigating this, I noticed something. Even number greater than 2 and a close relationship between prime numbers.
              I can not express what it is. but I can show you. that 'Goldbach' s conjecture can be used in the proof?
              With this method, I found the formula for the sum of a series of prime numbers in total, the two parameters you change the series instead of 2 greater than the total number of prime numbers, the sum gives couples.
              This parameter is reduced by generating equation can, until the total number N of primes can find, can I obtain a short equation. This equation will be deterministic, but the equation did not reduce.
              I wonder how can I do?
              In the meantime, while trying to reduce equation, I found many features related to prime numbers. One side is a parabola triangle, for example, in a geometric area of prime numbers has been collected, 6n +1, 6n-1 in the prime numbers reveals why he like.
            • djbroadhurst
              The operation of the law of small numbers in http://www.naturalsciences.be/expo/old_ishango/en/ishango/riddle.html reminds me of some messages on this list.
              Message 6 of 10 , Dec 2, 2009
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                The operation of the law of small numbers in
                http://www.naturalsciences.be/expo/old_ishango/en/ishango/riddle.html
                reminds me of some messages on this list.

                David
              • Paul Leyland
                ... Sorry for the late response to this thread but I ve been rather tied up with Real Life(tm) recently. There is a persuasive suggestion that the divisibility
                Message 7 of 10 , Dec 23, 2009
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                  On Tue, 2009-12-01 at 23:54 +0000, Phil Carmody wrote:

                  > > > published by Eratosthenes some 2200 years ago,
                  > > > and was certainly known some 1000
                  > > > (maybe 25 000) years earlier.
                  > >
                  > > I'm intrigued by the "certainly";
                  > > I would have said "probably" for 1k BCE.
                  >
                  > I'd have said "definitely" for >3k BCE. Base 60 just screams
                  > knowledge of divisibility properties.

                  Sorry for the late response to this thread but I've been rather tied up
                  with Real Life(tm) recently.

                  There is a persuasive suggestion that the divisibility properties of
                  radix-60 arithmetic is a consequence of its choice, not a reason for its
                  choice. The argument goes as follows.

                  A number of cultures have independently invented quinary arithmetic, for
                  reasons which should be obvious. There are still relics of this in
                  modern culture --- the five-bar-gate tallying method, for instance.
                  Bi-quinary has also been widely used throughout history. This uses four
                  different symbols for the digits 1-4 (the symbols are frequently 1 to 4
                  identical lines or dots) and another symbol for 5. Digits 6 through 9
                  are then represented by the juxtaposition of the 5-symbol and the
                  appropriate symbol for 1 through 4.

                  A number of cultures have independently invented duodecimal arithmetic.
                  Many relics of this exist: 12 ounces to the Troy pound; 12 inches to the
                  foot; 12 pennies to the shilling and so on. The most convincing
                  survivors to my mind are the survival of the English words "dozen" and
                  "gross".

                  Some time around 4000 to 3500 BCE the Sumerians moved into Mesopotamia
                  and merged with a pre-existing culture. One culture used quinary or
                  bi-quinary and the other duodecimal. Neither culture supplanted the
                  other, rather their notations merged. Indeed, the symbols of early
                  Mesopotamian arithmetic and accounting documents show strong evidence
                  for a bi-quinary (later decimal) sub-structure in the sexagesimal
                  notation.

                  If need be, I'll try and dig up the references from my catastrophically
                  disorganized library.

                  Paul
                • djbroadhurst
                  ... I learnt to count like that in Chi-Nyanja: modzi : one wiri : two tatu : three nai : four sanu :
                  Message 8 of 10 , Dec 23, 2009
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                    --- In primenumbers@yahoogroups.com,
                    Paul Leyland <paul@...> wrote:

                    > Digits 6 through 9 are then represented by the juxtaposition
                    > of the 5-symbol and the appropriate symbol for 1 through 4.

                    I learnt to count like that in Chi-Nyanja:

                    modzi : one
                    wiri : two
                    tatu : three
                    nai : four
                    sanu : hand
                    sanu ndi modzi : hand-and-one
                    sanu ndi wiri : hand-and-two
                    sanu ndi tatu : hand-and-three
                    sanu ndi nai : hand and-four
                    khumi : all-together

                    It seemed much more sensible than counting in French :-)

                    David
                  • Phil Carmody
                    ... That looks like the choice of 60 precisely because of its divisibility properties. They didn t take the LCM and later make a shock discovery that it had
                    Message 9 of 10 , Jan 3, 2010
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                      --- On Wed, 12/23/09, Paul Leyland <paul@...> wrote:
                      > On Tue, 2009-12-01 at 23:54 +0000, Phil Carmody wrote:
                      > > > > published by Eratosthenes some 2200 years ago,
                      > > > > and was certainly known some 1000
                      > > > > (maybe 25 000) years earlier.
                      > > >
                      > > > I'm intrigued by the "certainly";
                      > > > I would have said "probably" for 1k BCE.
                      > >
                      > > I'd have said "definitely" for >3k BCE. Base 60 just screams
                      > > knowledge of divisibility properties.
                      >
                      > Sorry for the late response to this thread but I've been
                      > rather tied up
                      > with Real Life(tm) recently.
                      >
                      > There is a persuasive suggestion that the divisibility
                      > properties of
                      > radix-60 arithmetic is a consequence of its choice, not a
                      > reason for its
                      > choice.  The argument goes as follows.
                      >
                      > A number of cultures have independently invented quinary
                      > arithmetic, for
                      > reasons which should be obvious.  There are still
                      > relics of this in
                      > modern culture --- the five-bar-gate tallying method, for
                      > instance.
                      > Bi-quinary has also been widely used throughout
                      > history.  This uses four
                      > different symbols for the digits 1-4 (the symbols are
                      > frequently 1 to 4
                      > identical lines or dots) and another symbol for 5. 
                      > Digits 6 through 9
                      > are then represented by the juxtaposition of the 5-symbol
                      > and the
                      > appropriate symbol for 1 through 4.
                      >
                      > A number of cultures have independently invented duodecimal
                      > arithmetic.
                      > Many relics of this exist: 12 ounces to the Troy pound; 12
                      > inches to the
                      > foot; 12 pennies to the shilling and so on.  The most
                      > convincing
                      > survivors to my mind are the survival of the English words
                      > "dozen" and
                      > "gross".
                      >
                      > Some time around 4000 to 3500 BCE the Sumerians moved into
                      > Mesopotamia
                      > and merged with a pre-existing culture.  One culture
                      > used quinary or
                      > bi-quinary and the other
                      > duodecimal.   Neither culture supplanted the
                      > other, rather their notations
                      > merged.   Indeed, the symbols of early
                      > Mesopotamian arithmetic and accounting documents show
                      > strong evidence
                      > for a bi-quinary (later decimal) sub-structure in the
                      > sexagesimal
                      > notation.

                      That looks like the choice of 60 precisely because of its divisibility properties. They didn't take the LCM and later make a shock discovery that it had all the factors of the two original numbers, shall we say.

                      Phil
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