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21202Re: [PrimeNumbers] Re: Set of prime numbers

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  • Paul Leyland
    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
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