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#1 2009-09-04 12:38:54

tony123
Member
Registered: 2007-08-03
Posts: 189

Find all pairs

Find all pairs (a,b) of integers

such that


Last edited by tony123 (2009-09-04 12:39:25)

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#2 2009-09-05 13:25:04

Fruityloop
Member
Registered: 2009-05-18
Posts: 120

Re: Find all pairs

I don't believe there are any integers because...









Last edited by Fruityloop (2009-09-05 13:32:52)


The eclipses from Algol (an eclipsing binary star) come further apart in time when the Earth is moving away from Algol and closer together in time when the Earth is moving towards Algol, thereby proving that the speed of light is variable and that Einstein's Special Theory of Relativity is wrong.

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#3 2009-09-05 17:24:19

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,460

Re: Find all pairs

Hi;

There are at least 2 solutions in Z

a=18 , b=3
a=3  ,  b=18

There may be more...


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#4 2009-09-06 00:46:48

Fruityloop
Member
Registered: 2009-05-18
Posts: 120

Re: Find all pairs

Good job Bobbym!

Oops...boy was I wrong.
(a+b)(a^2-ab+b^2)=3(2007-ab) doesn't mean (a+b)=3 or (a^2-ab+b^2)=3 shame

Last edited by Fruityloop (2009-09-06 00:54:52)


The eclipses from Algol (an eclipsing binary star) come further apart in time when the Earth is moving away from Algol and closer together in time when the Earth is moving towards Algol, thereby proving that the speed of light is variable and that Einstein's Special Theory of Relativity is wrong.

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#5 2009-09-06 01:11:47

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,460

Re: Find all pairs

Hi Fruityloop;

I am stuck on this one too.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#6 2009-10-27 05:20:20

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,460

Re: Find all pairs

Hi Tony123;

Prove:

That shows that 6021 can be factored into an integer square and another term. Also there are no negative solutions (If b were negative b^2 would of course be positive)

We could either try all 8 factors 2 at a time or notice that one factor has to be a square. Only 1 and 9 are possible. 1 doesn't work but 9 does. b^2 = 9 -> b=3.
The only solutions are:

3,18 and 18,3.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

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