- On Thu, 2012-02-02 at 19:42 +0000, Chris Caldwell wrote:
> I have a couple undergraduate students researching the history of the

To the Pythagoreans, primality and irrationality were closely related

> primality of one. For example, most of the early Greeks did not

> consider one to be a number, so one could not be a prime number for

> them. (A few considered primeness a subcategory of oddness, so two

> wasn't prime either!) As we move forward to the middle ages and

> later it is quite a mixture. For Cataldi, Euler, Gauss, and Landau,

> one appears not to be a prime. For Goldbach, Lebesgue, and Lehmer, it

> was a prime.

--- so closely related that they were almost identical concepts.

Although odd by modern standards, it arose from their fundamentally

geometric viewpoint and, in particular, from the concept of

measurability. To the Greeks, a number was necessarily greater than

one. Most on this later.

Think of a unit as being an unmarked ruler. A prime number is something

which can be measured only by a unit but is immeasurable by any other

number. A composite can be measured not only by a unit but also by

other numbers. This view is actually rather close to the modern

definition of a prime.

A rational is a length which may be measured by a unit if it is first

multiplied (i.e. multiple copies of the rational are placed end to end)

by a number.

The Greeks' concept of number makes good linguistic sense and tallies

quite well with modern English language. When we speak of "a number of

objects" or "a number of occurrences", we almost invariably refer to

more than two of them. One is not a number in this linguistic sense and

English, in common with most other languages, distinguishes between

singular and plural in a way which is both fundamental and pervasive.

That last statement also indicates why two is not really a number

either. English doesn't have much of the dual case left, but it still

distinguishes between one, two and many in constructs such as the

comparative and superlative, and the use of words and phrases such as

"either this or that but not both" and "among the options are".

Fascinating stuff if you like the history of the development of

intellectual activities.

Paul - --- In primenumbers@yahoogroups.com,

Phil Carmody <thefatphil@...> wrote:

> And we can't really decide whether we want companies

Is/are Manchester ******* United singular/plural?

> or bands to be singular or plural

Meanwhile, back in the archives: on page 252 of

http://www.gutenberg.org/ebooks/31246.html

Rouse Ball indicates that Mersenne may

have condered 2^p - 1 to be prime for p = 1:

"In the preface to the Cogitata a statement is made about

perfect numbers, which implies that the only values of p not

greater than 257 which make N prime, where N = 2^p - 1, are

1, 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257.."

However, Chris's students should check the original Latin for this.

So far they are expected to be adept in Greek, Latin, Italian,

German, French and English. Maybe Euler wrote something

relevant in Russian:

http://books.google.co.uk/books?id=cJTkQxTvGa4C

David