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prime and consecutive numbers

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  • fajriya hakim
    hi, except 2, every prime number is always sum of consecutive integer or every prime is always staircase numbers 3, = 1 + 2 5, = 2 + 3 7, = 3 + 4 11, = 5 +
    Message 1 of 7 , Dec 3, 2003
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      hi,
      except 2, every prime number is always sum of
      consecutive integer or
      every prime is always staircase numbers
      3, = 1 + 2
      5, = 2 + 3
      7, = 3 + 4
      11, = 5 + 6
      13, = 6 + 7
      17, = 8 + 9
      19, = 9 + 10
      23, = 11 + 12
      29, = 14 + 15
      31, = 15 + 16
      37, = 18 + 19
      41, = 20 + 21
      43, = 21 + 22
      47, = 23 + 24
      53, = 26 + 27
      59, = 29 + 30
      ...

      any idea ?

      > --- In primenumbers@yahoogroups.com, "piardapuka"
      > <singhns@h...>
      > wrote:
      > > hi to you all-im looking at the above and trying
      > to find various
      > > patterns-can anyone help
      > > piar
      >



      Fajriya Hakim

      Ph.D Student
      Institute for Mathematical Research (INSPEM)
      Universiti Putra Malaysia
      Telp. 60 3 8946 6879

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    • Paul Leyland
      ... After 2, all primes are odd and so of the form 2i+1. Your observation is then trivial. (HIBT?) Paul
      Message 2 of 7 , Dec 3, 2003
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        > except 2, every prime number is always sum of
        > consecutive integer or
        > every prime is always staircase numbers
        > 3, = 1 + 2
        > ...
        >
        > any idea ?

        After 2, all primes are odd and so of the form 2i+1. Your observation is then
        trivial.

        (HIBT?)


        Paul
      • Décio Luiz Gazzoni Filho
        ... Hash: SHA1 That is fairly obvious, and is valid not only for primes but in fact for any odd number, since such numbers can be written as 2n+1, which can be
        Message 3 of 7 , Dec 3, 2003
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          That is fairly obvious, and is valid not only for primes but in fact for any
          odd number, since such numbers can be written as 2n+1, which can be
          decomposed as n + (n+1) as claimed.

          Décio

          On Wednesday 03 December 2003 10:07, fajriya hakim wrote:
          > hi,
          > except 2, every prime number is always sum of
          > consecutive integer or
          > every prime is always staircase numbers
          > 3, = 1 + 2
          > 5, = 2 + 3
          > 7, = 3 + 4
          > 11, = 5 + 6
          > 13, = 6 + 7
          > 17, = 8 + 9
          > 19, = 9 + 10
          > 23, = 11 + 12
          > 29, = 14 + 15
          > 31, = 15 + 16
          > 37, = 18 + 19
          > 41, = 20 + 21
          > 43, = 21 + 22
          > 47, = 23 + 24
          > 53, = 26 + 27
          > 59, = 29 + 30
          > ...
          >
          > any idea ?
          >
          > > --- In primenumbers@yahoogroups.com, "piardapuka"
          > > <singhns@h...>
          > >
          > > wrote:
          > > > hi to you all-im looking at the above and trying
          > >
          > > to find various
          > >
          > > > patterns-can anyone help
          > > > piar
          >
          > Fajriya Hakim
          >
          > Ph.D Student
          > Institute for Mathematical Research (INSPEM)
          > Universiti Putra Malaysia
          > Telp. 60 3 8946 6879
          >
          > ---------------------------------
          > Do you Yahoo!?
          > Free Pop-Up Blocker - Get it now
          >
          > [Non-text portions of this message have been removed]
          >
          >
          >
          > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
          > The Prime Pages : http://www.primepages.org/
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        • Andy Swallow
          But every odd number is the sum of two consecutive numbers. It ain t a property reserved for primes... Andy
          Message 4 of 7 , Dec 4, 2003
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            But every odd number is the sum of two consecutive numbers. It ain't a
            property reserved for primes...

            Andy


            On Wed, Dec 03, 2003 at 04:07:41AM -0800, fajriya hakim wrote:
            > hi,
            > except 2, every prime number is always sum of
            > consecutive integer or
            > every prime is always staircase numbers
            > 3, = 1 + 2
            > 5, = 2 + 3
            > etc.
          • fajriyahakim
            after 2 all primes are odd, but odd numbers have 2 or more of sum consecutive numbers, for example, 9 = 4 + 5 and 9 = 2 + 3 + 4 15 = 7 + 8 and 15 = 4 + 5 + 6,
            Message 5 of 7 , Dec 5, 2003
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              after 2 all primes are odd, but odd numbers have 2 or more of sum
              consecutive numbers, for example,

              9 = 4 + 5 and 9 = 2 + 3 + 4
              15 = 7 + 8 and 15 = 4 + 5 + 6, 15 = 1 + 2 + 3 + 4 + 5

              but, prime number only can be built by 2 consecutive number
              is that right?
              if it is right, maybe the first prime number (2), have important
              meaning.



              --- In primenumbers@yahoogroups.com, "Paul Leyland" <pleyland@m...>
              wrote:
              > > except 2, every prime number is always sum of
              > > consecutive integer or
              > > every prime is always staircase numbers
              > > 3, = 1 + 2
              > > ...
              > >
              > > any idea ?
              >
              > After 2, all primes are odd and so of the form 2i+1. Your
              observation is then
              > trivial.
              >
              > (HIBT?)
              >
              >
              > Paul
            • Paul Jobling
              ... The sum of three consecutive numbers is always divisible by 3. The sum of five consecutive numbers is always divisible by 5. etc.
              Message 6 of 7 , Dec 5, 2003
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                > after 2 all primes are odd, but odd numbers have 2 or more of sum
                > consecutive numbers, for example,
                >
                > 9 = 4 + 5 and 9 = 2 + 3 + 4
                > 15 = 7 + 8 and 15 = 4 + 5 + 6, 15 = 1 + 2 + 3 + 4 + 5
                >
                > but, prime number only can be built by 2 consecutive number
                > is that right?

                The sum of three consecutive numbers is always divisible by 3.
                The sum of five consecutive numbers is always divisible by 5.
                etc.

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              • Paul Leyland
                ... Again, trivially so. The sum of all integers from 1 to a inclusive is given by a(a-1)/2. Therefore, the sum of all integers from b+1 to a inclusive is
                Message 7 of 7 , Dec 5, 2003
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                  > 9 = 4 + 5 and 9 = 2 + 3 + 4
                  > 15 = 7 + 8 and 15 = 4 + 5 + 6, 15 = 1 + 2 + 3 + 4 + 5
                  >
                  > but, prime number only can be built by 2 consecutive number
                  > is that right?

                  Again, trivially so. The sum of all integers from 1 to a
                  inclusive is given by a(a-1)/2.

                  Therefore, the sum of all integers from b+1 to a inclusive is
                  just a(a-1)/2 - b(b-1)/2. This expands to (a^2-a-b^2+b)/2
                  which factors to (a-b)(a+b-1)/2 which can only be prime if
                  (a-b)/2 = 1.


                  Paul
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