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Re: [PrimeNumbers] The history of the primality of one

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  • 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 1 of 9 , Feb 2, 2012
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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 ?
      >
      >
      >
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    • Paul Leyland
      ... To the Pythagoreans, primality and irrationality were closely related ... Although odd by modern standards, it arose from their fundamentally geometric
      Message 2 of 9 , Feb 3, 2012
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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 3 of 9 , Feb 3, 2012
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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 4 of 9 , Feb 3, 2012
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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 5 of 9 , Feb 3, 2012
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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 6 of 9 , Feb 3, 2012
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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 7 of 9 , Feb 3, 2012
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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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