21202Re: [PrimeNumbers] Re: Set of prime numbers
- Dec 23, 2009On Tue, 2009-12-01 at 23:54 +0000, Phil Carmody wrote:
> > > published by Eratosthenes some 2200 years ago,Sorry for the late response to this thread but I've been rather tied up
> > > 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.
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
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
If need be, I'll try and dig up the references from my catastrophically
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