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## Re: [PrimeNumbers] prime and consecutive numbers

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• 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.
• 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
• ... 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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• ... 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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