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Re: [PrimeNumbers] prime and consecutive numbers

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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 1 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 2 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 3 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 4 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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