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February 16th, 2011, 07:23 AM   #1
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Help needed - UFD

Ok, so I came across this problem and I think it is too difficult, so I'm asking for you guys help!

the problem is to find integer solutions to

first of all three solutions are n =1, 2, 13, I don't know if there is any other, probably not

lets factorize the whole thing like this:



is a UFD (http://en.wikipedia.org/wiki/Stark%E2%8 ... er_theorem), so we have



from here I'm not sure how to proceed exactly, but I think it is clear that for n>1 one of must be reductible, but it gives too many cases to check

another possibility is to write

Do the same the same argument of UFD holds? If yes, I think , yielding:

and where

I'm not sure how to proceed with the second factorization since is clearly irrational, do you guys see any mistake?

thanks in advance!
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February 16th, 2011, 10:32 AM   #2
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Re: Help needed - UFD

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Originally Posted by al-mahed
I'm afraid that I don't understand this conclusion. Are you saying that if ab = cd in a UFD, then gcd(a,b) = gcd(c,d)? That's not true in the rational integers: (6)(15) = (5)(1, but gcd(6,15) = 3 and gcd(5,1 = 1.
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February 16th, 2011, 10:45 AM   #3
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Re: Help needed - UFD

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Originally Posted by al-mahed
This isn't true in generality. in the Gaussian integers just like in the integers, but could be any divisor of 2 depending on the value of x.

LaTeX note: You don't need $ inside the tags, they're in math mode already. Also, the code for gcd is \gcd, not gcd.
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February 16th, 2011, 01:50 PM   #4
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Re: Help needed - UFD

Hmmmm, but by suposing there is a prime dividing both we get a contradiction, right? we get ... I see, there is one more condition, I need to take the Norms and see what divides what and if it is contradictory, I supose.
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February 16th, 2011, 02:04 PM   #5
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Re: Help needed - UFD

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Hmmmm, but by suposing there is a prime dividing both we get a contradiction, right?
No. If x = i, then gcd(i - i, i + i) = 2 (up to associates, of course; you could just as easily say 2i or -2). Can you find one that gives 1 - i? I can't immediately see one (though I haven't looked hard) nor can I immediately prove it does not occur.
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February 16th, 2011, 02:31 PM   #6
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Re: Help needed - UFD

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Quote:
Originally Posted by al-mahed
Hmmmm, but by suposing there is a prime dividing both we get a contradiction, right?
No. If x = i, then gcd(i - i, i + i) = 2 (up to associates, of course; you could just as easily say 2i or -2). Can you find one that gives 1 - i? I can't immediately see one (though I haven't looked hard) nor can I immediately prove it does not occur.
but we need to consider x integer

to the second question, you mean gcd=1-i? if yes
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