- --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
>

I had similar ideas as Phil on this ...

>

>

> --- 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";

It seems that the first estimation of about 9k BC has been revised to ~ 25k (Wikipedia says 20k, elsewhere I found 25k)

> 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?

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 - --- On Wed, 12/23/09, Paul Leyland <paul@...> wrote:
> On Tue, 2009-12-01 at 23:54 +0000, Phil Carmody wrote:

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.

> > > > 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.

Phil