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 Post subject: Equation in positive rationalsPosted: Wed, 2 Dec 2009 00:25:58 UTC
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Joined: Sun, 23 Mar 2008 10:55:47 UTC
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Solve the equation in the set of the positive rational numbers .

(It's easy to find the solutions in the set of the positive integer numbers).

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 Post subject: Posted: Wed, 2 Dec 2009 01:09:57 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 15563
Location: Austin, TX
?

This seems odd,

If I write my two rational numbers

in reduced form as

then there's a problem (one I'm certain is a consequence of GauÃŸ' lemma, but am too lazy to check).

since we're dealing with the positive integers, we can assume the raising to the is 1-1.

Then I raise both sides to the power. (ignore as that clearly gives a trivial result)

I get:

.

Now , so this means that our fraction is also a natural number. But then as we chose reduced, this implies , wherein was secretly a positive integer named in the first place.

The argument is symmetric in showing , so there are no extra rational answers, only the integer ones.

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 Post subject: Posted: Wed, 2 Dec 2009 17:59:50 UTC
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Joined: Sun, 23 Mar 2008 10:55:47 UTC
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What about the values and ??

Look carefully! There are many rational solutions and you can find all of them!

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 Post subject: Posted: Wed, 2 Dec 2009 20:49:02 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
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mathemagics wrote:
What about the values and ??

Look carefully! There are many rational solutions and you can find all of them!

Okay, I believe that. I made the mistake of concluding that the second term in my product was an integer when that's clearly crazy talk.

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 Post subject: Posted: Thu, 3 Dec 2009 04:20:46 UTC
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