- --- In primenumbers@yahoogroups.com, "murat.cagliyan" :

> A = (1,2,3,4, ... N) and B = (2,3,4, ... N) get. Number N of

Of course, A.A contains 1.A = { 1,..., N }

> up to a set of primes P,

> P = (A x A) / (B x B) / (1) shows.

> The number of primes up to here sum of N also can be found easily.

and B.B contains all composite numbers up to N, and

{ primes } = { all numbers } \ { 1, composite numbers }.

But your method has already been published by Erastothenes some 2200 years ago, and was certainly known some 1000 (maybe 25 000) years earlier.

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