February 16th, 2011, 07:23 AM  #1 
Senior Member Joined: Dec 2007 Posts: 687 Thanks: 47  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! 
February 16th, 2011, 10:32 AM  #2  
Senior Member Joined: Nov 2010 From: Berkeley, CA Posts: 174 Thanks: 35 Math Focus: Elementary Number Theory, Algebraic NT, Analytic NT  Re: Help needed  UFD Quote:
 
February 16th, 2011, 10:45 AM  #3  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Help needed  UFD Quote:
LaTeX note: You don't need $ inside the tags, they're in math mode already. Also, the code for gcd is \gcd, not gcd.  
February 16th, 2011, 01:50 PM  #4 
Senior Member Joined: Dec 2007 Posts: 687 Thanks: 47  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.

February 16th, 2011, 02:04 PM  #5  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Help needed  UFD Quote:
 
February 16th, 2011, 02:31 PM  #6  
Senior Member Joined: Dec 2007 Posts: 687 Thanks: 47  Re: Help needed  UFD Quote:
to the second question, you mean gcd=1i? if yes  

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