## prime and consecutive numbers

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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 +
Message 1 of 7 , Dec 3 4:07 AM
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 ?

> <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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[Non-text portions of this message have been removed]
• ... 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 4:22 AM
> 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
• ... 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 4:28 AM
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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
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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• 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 2:08 AM
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 5 of 7 , Dec 5 7:55 AM
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 6 of 7 , Dec 5 8:07 AM
> 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 7 of 7 , Dec 5 9:12 AM
> 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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