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The history of the primality of one

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  • Chris Caldwell
    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 11:42 AM
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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 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]
    • Walter Nissen
      A most interesting subject . I think the most accessible definition of prime is geometrical , rectangularization forces linearization ,
      Message 2 of 9 , Feb 2 12:41 PM
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        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 ?
      • Jack Brennen
        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 1:44 PM
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          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/
          >
          > Yahoo! Groups Links
          >
          >
          >
          >
          >
        • Paul Leyland
          ... 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 4:22 AM
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            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
          • djbroadhurst
            ... 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 9:26 AM
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              --- In primenumbers@yahoogroups.com,
              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
              http://de.academic.ru/dic.nsf/meyers/111736

              Des goûts et des couleurs, on ne dispute pas?

              David
            • djbroadhurst
              ... 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 1:28 PM
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                --- In primenumbers@yahoogroups.com,
                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:
                http://books.google.co.uk/books?id=ea8WAAAAQAAJ

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

                David
              • djbroadhurst
                ... Here is a thumbnail biography: http://www.les-mathematiques.net/phorum/read.php?17,323622 ... David
                Message 7 of 9 , Feb 3 1:38 PM
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                  --- In primenumbers@yahoogroups.com,
                  "djbroadhurst" <d.broadhurst@...> wrote:

                  > V. A. Lebesgue designated 1 as prime on page 5 of his
                  > 1859 textbook:
                  > http://books.google.co.uk/books?id=ea8WAAAAQAAJ

                  Here is a thumbnail biography:
                  http://www.les-mathematiques.net/phorum/read.php?17,323622
                  > Un mathématicien méconnu aujourd'hui mais qui a
                  > joué un rôle important dans la première moitié du XIX ème.

                  David
                • Phil Carmody
                  ... 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 5:24 PM
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                    --- 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?!?!)
                  • djbroadhurst
                    ... 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 5:48 PM
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                      --- In primenumbers@yahoogroups.com,
                      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:

                      "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
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