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21251Re: [PrimeNumbers] Re: Set of prime numbers

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  • Phil Carmody
    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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