Math Is Fun Forum

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,428

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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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

mathsyperson

Moderator

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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It wanted to be normal.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,428

Re: I need a mathematical proof

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

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

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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Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: I need a mathematical proof

Should be a simple proof by induction.

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"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..."

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: I need a mathematical proof

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

will do

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X'(y-Xβ)=0

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

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

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X'(y-Xβ)=0

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,428

Re: I need a mathematical proof

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

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

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?

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"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..."

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,428

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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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

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?

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.

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X'(y-Xβ)=0