- 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, 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

observation is then

> > 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

> trivial.

>

> (HIBT?)

>

>

> Paul > after 2 all primes are odd, but odd numbers have 2 or more of sum

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

> 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 five consecutive numbers is always divisible by 5.

etc.

__________________________________________________

Virus checked by MessageLabs Virus Control Centre.> 9 = 4 + 5 and 9 = 2 + 3 + 4

Again, trivially so. The sum of all integers from 1 to a

> 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?

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