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RE: [PrimeNumbers] prime and consecutive numbers

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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 1 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 2 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
        >
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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 3 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 4 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 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?

              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 6 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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