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

ganesh
Moderator
Registered: 2005-06-28
Posts: 20,747

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.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

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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: 20,747

Re: I need a mathematical proof

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


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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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
Member
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
Moderator
Registered: 2005-06-28
Posts: 20,747

Re: I need a mathematical proof

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


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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

Ricky
Moderator
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: 20,747

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.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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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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