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 Post subject: Tricky cubic (Number Theory)Posted: Wed, 22 Feb 2012 03:30:30 UTC
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For what integers is the equation satisfied?

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 Post subject: Re: Tricky cubic (Number Theory)Posted: Wed, 22 Feb 2012 03:55:29 UTC
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For what integers is the equation satisfied?

I know this is not what you are looking for, but and work.
As far as a general solution, I have no idea.

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 Post subject: Re: Tricky cubic (Number Theory)Posted: Wed, 22 Feb 2012 03:56:50 UTC
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Justin wrote:
For what integers is the equation satisfied?

I know this is not what you are looking for, but and work.
As far as a general solution, I have no idea.

I'll give you a hint: there are only finitely many solutions and more than you have already posted.

Also, note you have a square for y, so you may as well add 5 and -11 to up your nonexistent "score"

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 Post subject: Re: Tricky cubic (Number Theory)Posted: Wed, 22 Feb 2012 04:25:27 UTC
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Another hint:
Spoiler:
is Euclidean, and all units are cubes.

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 Post subject: Re: Tricky cubic (Number Theory)Posted: Wed, 22 Feb 2012 04:55:44 UTC
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Spoiler:
Hmmm...I didn't think about using Gaussian integers.
I don't get the hint about "units being cubes" though.
I will have to ponder this some more...obviously, you are prompting me
to use some clever factorization trick here(?)

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 Post subject: Re: Tricky cubic (Number Theory)Posted: Wed, 22 Feb 2012 05:22:51 UTC
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That is to absorb the unit(s) that arise when you invoke unique factorisation.

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 Post subject: Re: Tricky cubic (Number Theory)Posted: Wed, 22 Feb 2012 15:20:30 UTC
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Justin: You'll find that most diophantine equations require at least some machinery from something a bit more potent than elementary methods to solve as completely as necessary. This one is about as simple as it gets without being essentially trivial, which is why I chose it, since it's still a bit involved.

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