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

Find all pairs (a,b) of integers

such that

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

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**Fruityloop****Member**- Registered: 2009-05-18
- Posts: 132

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

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

**The eclipses from Algol 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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 103,717

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.****If it ain't broke, fix it until it is.** **A number by itself is useful, but it is far more useful to know how accurate or certain that number is.**

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**Fruityloop****Member**- Registered: 2009-05-18
- Posts: 132

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

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

**The eclipses from Algol 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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 103,717

Hi Fruityloop;

I am stuck on this one too.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.** **A number by itself is useful, but it is far more useful to know how accurate or certain that number is.**

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 103,717

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.****If it ain't broke, fix it until it is.** **A number by itself is useful, but it is far more useful to know how accurate or certain that number is.**

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