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#1 2006-09-29 00:35:08

ganesh
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Registered: 2005-06-28
Posts: 14,803

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 00:50:28

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

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 2006-10-06 02:48:35

ganesh
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Registered: 2005-06-28
Posts: 14,803

Re: I need a mathematical proof

The only condition is a,b belong to Natural numbers.
Thanks mathsyperson.


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#4 2006-10-06 07:33:11

espeon
Real Member
Registered: 2006-02-05
Posts: 2,586

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 2006-10-06 07:57:59

Ricky
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Registered: 2005-12-04
Posts: 3,791

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..."

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#6 2006-10-06 19:10:44

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

Re: I need a mathematical proof

Yes, but that doesn't need to be an induction.


will do


X'(y-Xβ)=0

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#7 2006-10-06 19:12:24

George,Y
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Registered: 2006-03-12
Posts: 1,306

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

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#8 2006-10-06 20:05:19

ganesh
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Registered: 2005-06-28
Posts: 14,803

Re: I need a mathematical proof

Thanks George and Ricky, I shall study the proof you have given.


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#9 2006-10-07 05:58:51

Ricky
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Registered: 2005-12-04
Posts: 3,791

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..."

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#10 2006-10-07 17:15:23

ganesh
Moderator
Registered: 2005-06-28
Posts: 14,803

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.


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#11 2006-10-08 17:35:17

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

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^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?

Sure, I shall illustrate Post #6 in detail.

Typically, we wanna know if a[sup]n[/sup]-b[sup]n[/sup] 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

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