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#1 2006-09-29 22:35:08
- ganesh
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I need a mathematical proof
Hi
Cany anyone of you, using mathematical induction or otherwise, give a flawless proof of (a^n-b^n) always being divisble by (a-b), if the statement is true?
I shall be glad even if someone gives a counter-proof.
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#2 2006-09-29 22:50:28
- mathsyperson
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Re: I need a mathematical proof
Are there any restrictions on what a, b and n can be?
Presumably a ≠b, but do they all have to be integers as well, or all rationals, or something?
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#3 2006-10-07 00:48:35
- ganesh
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Re: I need a mathematical proof
The only condition is a,b belong to Natural numbers.
Thanks mathsyperson.
Character is who you are when no one is looking.
#4 2006-10-07 05:33:11
- espeon
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Re: I need a mathematical proof
hmm if a=2 and b=3 and n=2 then 4-9=-5 which isnt divisible by a-b=-1 and aaaaaaaaaaaaaaaaaaaaaa im getting confused! *Gets dizzy* it would of worked if i knew if -5 was divisible by -1 lol
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#5 2006-10-07 05:57:59
- Ricky
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Re: I need a mathematical proof
Should be a simple proof by induction.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
#6 2006-10-07 17:10:44
- George,Y
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Re: I need a mathematical proof
Yes, but that doesn't need to be an induction.
will do
X'(y-Xβ)=0
#7 2006-10-07 17:12:24
- George,Y
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Re: I need a mathematical proof
for instance,
a(aa+ab+bb)-b(aa+ab+bb)= (a-b)(aa+ab+bb)= a^3-b^3
X'(y-Xβ)=0
#8 2006-10-07 18:05:19
- ganesh
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Re: I need a mathematical proof
Thanks George and Ricky, I shall study the proof you have given.
Character is who you are when no one is looking.
#9 2006-10-08 03:58:51
- Ricky
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Re: I need a mathematical proof
for instance,
a(aa+ab+bb)-b(aa+ab+bb)= (a-b)(aa+ab+bb)= a^3-b^3
You know better than anyone George that an instance does not make a proof.
I'm not sure how you would algebraically solve your formula, would you mind showing me the steps?
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
#10 2006-10-08 15:15:23
- ganesh
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Re: I need a mathematical proof
I am looking forward to a clear proof by mathematical induction, a flawless elegant proof. Ricky, George and mathsyperon, help me. Help from any other source is most welcome.
Character is who you are when no one is looking.
#11 2006-10-09 15:35:17
- George,Y
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Re: I need a mathematical proof
Ricky wrote:
for instance,
a(aa+ab+bb)-b(aa+ab+bb)= (a-b)(aa+ab+bb)= a^3-b^3You know better than anyone George that an instance does not make a proof.
I'm not sure how you would algebraically solve your formula, would you mind showing me the steps?
Sure, I shall illustrate Post #6 in detail.
Typically, we wanna know if an-bn could be expressed as (a-b)A, where A is some polynomial.
A could be find out-
Hence
and
Using notation, the proof would be:
Hence
t and k are indexes representing integars, so we can equate them when we do the following algebra.
X'(y-Xβ)=0