Author |
Topic: Expressibility (Read 8376 times) |
|
ThudnBlunder
wu::riddles Moderator
Uberpuzzler
The dewdrop slides into the shining Sea
Gender: 
Posts: 4489
|
 |
Expressibility
« on: Aug 29th, 2004, 10:54pm » |
 Quote  Modify
|
Which positive integers cannot be expressed as a sum of 2 or more consecutive integers?
|
|
 IP Logged |
THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
|
|
|
mistysakura
Junior Member
Gender: 
Posts: 121
|
|
Re: Expressibility
« Reply #1 on: Aug 30th, 2004, 4:15am » |
Quote Modify
|
none.
Let the number to be expressed be x
(-x+1)+(-x+2)...+(-1)+0+1...+(x-2)+(x-1)+x
=0+x
=x
(Exception: 1=0+1 (duh.))
|
| « Last Edit: Aug 30th, 2004, 4:17am by mistysakura » |
IP Logged |
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: Expressibility
« Reply #2 on: Aug 30th, 2004, 4:35am » |
Quote Modify
|
ok then, a 'follow-up question' 
Which positive integers cannot be expressed as a sum of 2 or more consecutive positive integers?
|
| « Last Edit: Aug 30th, 2004, 4:36am by towr » |
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
EZ_Lonny
Full Member
Real anarchists play chess without the king
Gender:
Posts: 168
|
|
Re: Expressibility
« Reply #3 on: Aug 30th, 2004, 8:19am » |
Quote Modify
|
Let me see ...........
At least all even Integers cannot be displayed, 'coz in two consecutive integers there is one odd and one even integer.
Even + odd = odd excuse me if i'm wrong
all odd integers can be made. If only i knew how to proof that.
1+2=3 ; 2+3=5; 3+4=7; 4+5=9 etc.
|
|
IP Logged |
There is much pleasure to be gained from useless knowledge - Bertrand Russel
|
|
|
BNC
Uberpuzzler
Gender:
Posts: 1732
|
|
Re: Expressibility
« Reply #4 on: Aug 30th, 2004, 8:22am » |
Quote Modify
|
on Aug 30th, 2004, 4:35am, towr wrote:ok then, a 'follow-up question'
Which positive integers cannot be expressed as a sum of 2 or more consecutive positive integers? |
|
Obviously all odd numbers are ok. The question is what even numbers may not be expressed that way.
|
| « Last Edit: Aug 30th, 2004, 8:23am by BNC » |
IP Logged |
How about supercalifragilisticexpialidociouspuzzler [Towr, 2007]
|
|
|
EZ_Lonny
Full Member
Real anarchists play chess without the king
Gender:
Posts: 168
|
|
Re: Expressibility
« Reply #5 on: Aug 30th, 2004, 8:22am » |
Quote Modify
|
I know, I'm working on that
|
|
IP Logged |
There is much pleasure to be gained from useless knowledge - Bertrand Russel
|
|
|
EZ_Lonny
Full Member
Real anarchists play chess without the king
Gender:
Posts: 168
|
|
Re: Expressibility
« Reply #6 on: Aug 30th, 2004, 8:24am » |
Quote Modify
|
I'm only guessing:
All integers dividable by 4 or that odd onez
|
|
IP Logged |
There is much pleasure to be gained from useless knowledge - Bertrand Russel
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: Expressibility
« Reply #7 on: Aug 30th, 2004, 9:15am » |
Quote Modify
|
2+3+4+5=14
So even some even numbers not divisble by 4 can be expressed by 2 or more consecutive numbers..
|
| « Last Edit: Aug 30th, 2004, 9:15am by towr » |
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
Sir Col
Uberpuzzler
impudens simia et macrologus profundus fabulae
Gender:
Posts: 1825
|
|
Re: Expressibility
« Reply #8 on: Aug 30th, 2004, 4:12pm » |
Quote Modify
|
In fact, all integers greater than 2 that are not divisible by 4 can be expressed as the sum of at least two positive consecutive integers. However, the proof of that statement requires a slightly different perspective. As towr suggested, instead of trying to work out which integers can be written as such a sum, try and work out which integers cannot...
on Aug 30th, 2004, 8:24am, EZ_Lonny wrote:I'm only guessing:
All integers dividable by 4 or that odd onez |
|
What about 8? Clearly some divisible by 4 work and some don't.
|
| « Last Edit: Aug 30th, 2004, 5:04pm by Sir Col » |
IP Logged |
mathschallenge.net / projecteuler.net
|
|
|
Hooie
Newbie
Gender:
Posts: 29
|
|
Re: Expressibility
« Reply #9 on: Aug 30th, 2004, 6:37pm » |
Quote Modify
|
I think I got it.
* It seems to be powers of 2 that don't work. Now I'm trying to find out why. I generalized a number formed by adding consecutive integers to the form of (n+1)x + (n/2)(n+1). I set that equal to 2^k and I'm messing around til I find a contradiction.*
Can someone tell me if I'm on the right track? Or am I just making this all up?
|
|
IP Logged |
|
|
|
Aryabhatta
Uberpuzzler
Gender:
Posts: 1321
|
|
Re: Expressibility
« Reply #10 on: Aug 30th, 2004, 6:41pm » |
Quote Modify
|
on Aug 30th, 2004, 6:37pm, Hooie wrote:I think I got it.
Can someone tell me if I'm on the right track? Or am I just making this all up? |
|
You are on the right track. (you have the right answer)
|
|
IP Logged |
|
|
|
Grimbal
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 7526
|
|
Re: Expressibility
« Reply #11 on: Aug 31st, 2004, 3:57am » |
Quote Modify
|
::
Let's write m = 2k*(2n+1)
If 2k > n then m can be written as
m = sum 2k-n ... 2k+n
there are 2n+1 terms averaging 2k.
If 2k <= n, then m can be written as
m = sum n-2k+1 ... n+2k
there are 2*2k terms averaging n+1/2 = (2n+1)/2
The first case is a valid solution when n>0, else there is only one term
The second case is always valid, there are always at lest 2 terms.
This proves that all numbers can be written as such a sum, except when n=0, which means m is a power of 2.
It remains to prove that powers of 2 can never be written in such a way.
Lets say m = sum a ... b is a power of 2
m = (a+b)*(b-a+1)/2, b>a, a>0.
2m = (a+b)*(b-a+1) must also be a power of 2.
But (a+b) or (b-a+1) must be odd. The only way out if if the odd factor is 1.
But a>0 => (a+b) >= 1+2 and b>a => (b-a+1)>=2, so it cannot be.
::
|
| « Last Edit: Aug 31st, 2004, 3:58am by Grimbal » |
IP Logged |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print