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Is this true?

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  • cashogor
    Hi! Let S(C,B) be the sum of the digits of C in base B Let mod(x,y) be the remainder of x/y (integer division) I´ve just found that: Sum (n=1 to oo) (
    Message 1 of 9 , Feb 14, 2002
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      Hi!

      Let S(C,B) be the sum of the digits of C in base B
      Let mod(x,y) be the remainder of x/y (integer division)

      I´ve just found that:

      Sum (n=1 to oo) ( mod(C,B^n) / B^n ) = S(C,B)/(B-1)

      I cannot prove it because I´m not too smart ;-)
      Does it sound familiar to anybody? Maybe it´s obvious.

      Thanks

      Néstor
    • cashogor
      Hi! ... Write C in base B as (c_{m-1} c_{m-2} ... c_ 1 c_0)_B . For n = m we have mod(C, B^n) = C but for n
      Message 2 of 9 , Feb 18, 2002
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        Hi!

        Robin Chapman wrote this on Sci.math. I think it is correct:

        ---------------------------------------------------------

        Write C in base B as (c_{m-1} c_{m-2} ... c_ 1 c_0)_B .
        For n >= m we have mod(C, B^n) = C but for n < m
        we have mod(C, B^n) = (c_{n-1} c_{n-2} ... c_ 1 c_0)_B
        = c_0 + c_1 B + ... + c_{n-1} B^{n-1}
        The sum is
        c_0/B
        + (c_0/B^2 + c_1/B)
        + (c_0/B^3 + c_1/B^2 + c_2/B)
        + ....
        + (c_0/B^m + c_1/B^{m-1} + ... + c_{m-1}/B)
        + ...
        + (c_0/B^n + c_1/B^{n-1} + ... + c_{m-1}/B^{m-m+1})
        + ...

        = (c_0 + c_1 + ... + c_{m-1})(1/B + 1/B^2 + ...)

        = (c_0 + c_1 + ... + c_{m-1})/(B-1).


        --------------------------------------------------------

        Thanks

        Néstor
      • Phil Carmody
        ... If Robin wrote it, then that s the default opinion you should start with. ... The sum of what? ... Why are you dividing by B? ... and by B^2? ... Certainly
        Message 3 of 9 , Feb 18, 2002
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          On Mon, 18 February 2002, "cashogor" wrote:
          > Hi!
          >
          > Robin Chapman wrote this on Sci.math. I think it is correct:

          If Robin wrote it, then that's the default opinion you should start with.

          > Write C in base B as (c_{m-1} c_{m-2} ... c_ 1 c_0)_B .
          > For n >= m we have mod(C, B^n) = C but for n < m
          > we have mod(C, B^n) = (c_{n-1} c_{n-2} ... c_ 1 c_0)_B
          > = c_0 + c_1 B + ... + c_{n-1} B^{n-1}
          > The sum is

          The sum of what?

          > c_0/B

          Why are you dividing by B?

          > + (c_0/B^2 + c_1/B)

          and by B^2?

          > + (c_0/B^3 + c_1/B^2 + c_2/B)
          > + ....
          > + (c_0/B^m + c_1/B^{m-1} + ... + c_{m-1}/B)
          > + ...
          > + (c_0/B^n + c_1/B^{n-1} + ... + c_{m-1}/B^{m-m+1})
          > + ...
          >
          > = (c_0 + c_1 + ... + c_{m-1})(1/B + 1/B^2 + ...)
          >
          > = (c_0 + c_1 + ... + c_{m-1})/(B-1).

          Certainly the right hand side is correctly manipulated, although the final step does require more rigour - it's not an infinite sum - but to be honest I can't see what the question, to which this is an answer, is.

          Phil

          Don't be fooled, CRC Press are _not_ the good guys.
          They've taken Wolfram's money - _don't_ give them yours.
          http://mathworld.wolfram.com/erics_commentary.html


          Find the best deals on the web at AltaVista Shopping!
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        • cashogor
          Hi Phil! ... Let S(C,B) be the sum of the digits of C in base B Let mod(x,y) be the remainder of x/y (integer division) I´ve just found that: Sum (n=1 to oo)
          Message 4 of 9 , Feb 18, 2002
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            Hi Phil!

            I posted a message recently that said:

            ---------------------------------------------------

            Let S(C,B) be the sum of the digits of C in base B
            Let mod(x,y) be the remainder of x/y (integer division)

            I´ve just found that:

            Sum (n=1 to oo) ( mod(C,B^n) / B^n ) = S(C,B)/(B-1)

            I cannot prove it because I´m not too smart ;-)
            Does it sound familiar to anybody? Maybe it´s obvious.

            -----------------------------------------------------

            And that´s what Robin seems to have proved.

            Néstor.
          • hamathib
            Point (1): There is a pattern in compound numbers. This implies that compound numbers have a function. Point (2): There is a relationship between compound
            Message 5 of 9 , Sep 2, 2009
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              Point (1): There is a pattern in compound numbers. This implies that compound numbers have a function.

              Point (2): There is a relationship between compound numbers and squares of primes. This implies that squares of primes have a function dependent on the function of compounds.

              Point (3): The pattern in prime numbers is related to "Point 2". This implies that prime numbers have a function.

              Regards.
            • Billy Hamathi
              Why do I make very silly mistakes even when I am serious, the question should read: Point (1): There is a pattern in composite numbers. This implies that
              Message 6 of 9 , Sep 2, 2009
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                Why do I make very silly mistakes even when I am serious, the question should read:

                Point (1): There is a pattern in composite numbers. This implies that composite numbers have a function.

                Point (2): There is a relationship between composite numbers and squares of primes. This implies that squares of primes have a function dependent on the function of composites.

                Point (3): The pattern in prime numbers is related to "Point 2". This implies that prime numbers have a function.




                [Non-text portions of this message have been removed]
              • Chris Caldwell
                I am not sure what your point is, but it seems to center on the Urban myths that there are no functions which describe the primes. There are many (none
                Message 7 of 9 , Sep 2, 2009
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                  I am not sure what your point is, but it seems to center on the Urban
                  myths that there are no functions which describe the primes. There are
                  many (none particularly useful, but dozens have been published). For
                  subsets of the primes, my favorite is Mills'

                  The primes are a pattern.

                  -----Original Message-----
                  From: primenumbers@yahoogroups.com [mailto:primenumbers@yahoogroups.com]
                  On Behalf Of Billy Hamathi
                  Sent: Wednesday, September 02, 2009 4:50 AM
                  To: primenumbers@yahoogroups.com
                  Subject: [PrimeNumbers] Re: Is this true?


                  Why do I make very silly mistakes even when I am serious, the question
                  should read:

                  Point (1): There is a pattern in composite numbers. This implies that
                  composite numbers have a function.

                  Point (2): There is a relationship between composite numbers and squares
                  of primes. This implies that squares of primes have a function dependent
                  on the function of composites.

                  Point (3): The pattern in prime numbers is related to "Point 2". This
                  implies that prime numbers have a function.




                  [Non-text portions of this message have been removed]



                  ------------------------------------

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                • Billy Hamathi
                  Thanks Chris, but could you answer me without answering what my question could imply? Is it true? ... From: Chris Caldwell Subject: RE:
                  Message 8 of 9 , Sep 2, 2009
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                    Thanks Chris, but could you answer me without answering what my question could imply? Is it true?

                    --- On Wed, 2/9/09, Chris Caldwell <caldwell@...> wrote:


                    From: Chris Caldwell <caldwell@...>
                    Subject: RE: [PrimeNumbers] Re: Is this true?
                    To: primenumbers@yahoogroups.com
                    Date: Wednesday, 2 September, 2009, 4:35 PM


                     



                    I am not sure what your point is, but it seems to center on the Urban
                    myths that there are no functions which describe the primes. There are
                    many (none particularly useful, but dozens have been published). For
                    subsets of the primes, my favorite is Mills'

                    The primes are a pattern.

                    -----Original Message-----
                    From: primenumbers@ yahoogroups. com [mailto:primenumbers@ yahoogroups. com]
                    On Behalf Of Billy Hamathi
                    Sent: Wednesday, September 02, 2009 4:50 AM
                    To: primenumbers@ yahoogroups. com
                    Subject: [PrimeNumbers] Re: Is this true?

                    Why do I make very silly mistakes even when I am serious, the question
                    should read:

                    Point (1): There is a pattern in composite numbers. This implies that
                    composite numbers have a function.

                    Point (2): There is a relationship between composite numbers and squares
                    of primes. This implies that squares of primes have a function dependent
                    on the function of composites.

                    Point (3): The pattern in prime numbers is related to "Point 2". This
                    implies that prime numbers have a function.

                    [Non-text portions of this message have been removed]

                    ------------ --------- --------- ------

                    Unsubscribe by an email to: primenumbers- unsubscribe@ yahoogroups. com
                    The Prime Pages : http://www.primepag es.org/

                    Yahoo! Groups Links



















                    [Non-text portions of this message have been removed]
                  • Yann Guidon
                    ... Do you have time to elaborate a bit on that ? Though I think that I know what you are talking about, I would like to read your own version, with examples
                    Message 9 of 9 , Sep 2, 2009
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                      Chris Caldwell wrote:
                      > I am not sure what your point is, but it seems to center on the Urban
                      > myths that there are no functions which describe the primes. There are
                      > many (none particularly useful, but dozens have been published). For
                      > subsets of the primes, my favorite is Mills'
                      >
                      > The primes are a pattern.

                      Do you have time to elaborate a bit on that ?
                      Though I think that I know what you are talking
                      about, I would like to read your own version,
                      with examples etc.

                      Best regards,
                      yg
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