## The history of the primality of one

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• I have a couple undergraduate students researching the history of the primality of one. For example, most of the early Greeks did not consider one to be a
Message 1 of 9 , Feb 2, 2012
I have a couple undergraduate students researching the history of the 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.

Here is my question: if you have any good references on what was believe by who in the past (preferably before 1900 and especially if it says why), please let me know off the list: caldwell@...<mailto:caldwell@...>. I would be glad to share what we have collected so far, but we are just beginning (for example, trying to find copies the sources cited in Dickson history...), so it might be far more productive to wait a bit (it will eventually be web page on the prime pages).

This is of course just a matter of definition, but definitions are motivated by their usage. Usage now demands that one be (a number,) a unit
and neither prime nor composite, but as the names above suggests, that was not always the case.

Chris Caldwell

[Non-text portions of this message have been removed]
• A most interesting subject . I think the most accessible definition of prime is geometrical , rectangularization forces linearization ,
Message 2 of 9 , Feb 2, 2012
A most interesting subject .

I think the most accessible definition of prime is geometrical ,
rectangularization forces linearization ,
http://upforthecount.com/math/nnnp1np1.html
Trivially , 1 is prime .

I'd be happy to live in a world where 1 is both prime and perfect .
If 1 is not perfect , then by what stretch of the imagination is
Euler's phi ( 1 ) = 1 ?
I find the equality in phi ( 1 ) = 1 jarring .
Are units the easiest or hardest part of algebra ?
• phi(1) has to equal 1 in order to preserve the multiplicative nature of phi(). In other words, If gcd(a,b) == 1, then phi(a)*phi(b)==phi(a*b). 1 is the only
Message 3 of 9 , Feb 2, 2012
phi(1) has to equal 1 in order to preserve the multiplicative
nature of phi(). In other words,

If gcd(a,b) == 1, then phi(a)*phi(b)==phi(a*b).

1 is the only value for phi(1) which preserves this property,
and the property is too important to give up.

Note that one definition of primality, which could include
negative primes, would be that abs(phi(X))+1 == abs(X) if
and only if X is prime. By such a definition, 1 would then
not be prime.

On 2/2/2012 12:41 PM, Walter Nissen wrote:
> A most interesting subject .
>
> I think the most accessible definition of prime is geometrical ,
> rectangularization forces linearization ,
> http://upforthecount.com/math/nnnp1np1.html
> Trivially , 1 is prime .
>
> I'd be happy to live in a world where 1 is both prime and perfect .
> If 1 is not perfect , then by what stretch of the imagination is
> Euler's phi ( 1 ) = 1 ?
> I find the equality in phi ( 1 ) = 1 jarring .
> Are units the easiest or hardest part of algebra ?
>
>
>
> ------------------------------------
>
> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
> The Prime Pages : http://www.primepages.org/
>
>
>
>
>
>
• ... To the Pythagoreans, primality and irrationality were closely related ... Although odd by modern standards, it arose from their fundamentally geometric
Message 4 of 9 , Feb 3, 2012
On Thu, 2012-02-02 at 19:42 +0000, Chris Caldwell wrote:
> I have a couple undergraduate students researching the history of the
> 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.

To the Pythagoreans, primality and irrationality were closely related
--- 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
• ... But then some Germans disagreed, circa 1900: http://www.peterhug.ch/lexikon/primzahl/13_0390 http://de.academic.ru/dic.nsf/meyers/111736 Des goûts et des
Message 5 of 9 , Feb 3, 2012
Paul Leyland <paul@...> wrote:

> The Greeks' concept of number makes good linguistic sense
> and tallies quite well with modern English language.

But then some Germans disagreed, circa 1900:

http://www.peterhug.ch/lexikon/primzahl/13_0390

Des goûts et des couleurs, on ne dispute pas?

David
• ... V. A. Lebesgue designated 1 as prime on page 5 of his 1859 textbook: http://books.google.co.uk/books?id=ea8WAAAAQAAJ He is not to be confused with Henri
Message 6 of 9 , Feb 3, 2012
Chris Caldwell <caldwell@...> wrote:

> For Goldbach, Lebesgue, and Lehmer, it was a prime.

V. A. Lebesgue designated 1 as prime on page 5 of his
1859 textbook:

He is not to be confused with Henri Lebesque,
who was not born until 1875.

David
• ... Here is a thumbnail biography: http://www.les-mathematiques.net/phorum/read.php?17,323622 ... David
Message 7 of 9 , Feb 3, 2012

> V. A. Lebesgue designated 1 as prime on page 5 of his
> 1859 textbook:

Here is a thumbnail biography:
> Un mathématicien méconnu aujourd'hui mais qui a
> joué un rôle important dans la première moitié du XIX ème.

David
• ... Fundamental, pervasive, and perverted. It is, after all, a language in which the singular thou has been jetisoned for the plural you , and similarly the
Message 8 of 9 , Feb 3, 2012
--- On Fri, 2/3/12, Paul Leyland wrote:
> 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.

Fundamental, pervasive, and perverted.

It is, after all, a language in which the singular 'thou' has
been jetisoned for the plural 'you', and similarly the plural
'they' adopted as a singular when trying to avoid mentioning
gender. And we can't really decide whether we want companies
or bands to be singular or plural - is Nokia going down the
pan, or are Nokia going down the pan? (Which does seem to
correlate strongly with pondianness.)

Phil
(who recently left a country where in "5 boys", "boys" is *not* plural?!?!)
• ... Is/are Manchester ******* United singular/plural? Meanwhile, back in the archives: on page 252 of http://www.gutenberg.org/ebooks/31246.html Rouse Ball
Message 9 of 9 , Feb 3, 2012
Phil Carmody <thefatphil@...> wrote:

> And we can't really decide whether we want companies
> or bands to be singular or plural

Is/are Manchester ******* United singular/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: