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Re: Set of prime numbers

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  • 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 1 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 2 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 3 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 4 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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