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Math Help - [SOLVED] Complete Square

  1. #16
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    Another proof idea

    Opalg, to continue the argument from your first post, that pa^2-qb^2=1,pc^2-qd^2=-1 has no solution for p,q coprime and squarefree...

    Given a,b,c,d, the solution is p=\frac{b^2+d^2}D,q=\frac{a^2+c^2}D, D=(ad)^2-(bc)^2 Can it be shown that D\not|gcd(b^2+d^2,a^2+c^2), and therefore no integer solutions to p,q exist unless D=1? I think you will be able to see that if D=1 then p or q must be 1, proving the theorem. I have gone only far enough to show a,d must be odd and b,c must be even, making D\equiv1\bmod4.
    Last edited by Media_Man; February 15th 2010 at 12:14 PM. Reason: typo
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  2. #17
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    A proof, at last

    Attached is a write-up of a final proof (it's three pages long). This problem was fascinatingly complicated. I knew virtually nothing about the Pell equation and continued fractions, so I have had the pleasant opportunity these last few days to understand how they work. Number theory really is enchanting the way a simple question can lead to an arbitrarily complex answer.

    havaliza: Where the hell did this question come from? A book? A class? Your own wanderings?

    After reading and understanding the attached proof, I wonder if anyone wants to look at a generalization involving x(y+n),y(x+n)...
    Attached Files Attached Files
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  3. #18
    Super Member Bacterius's Avatar
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    I'll save it on my computer and read it in my next free time.

    Number theory really is enchanting the way a simple question can lead to an arbitrarily complex answer.
    What the hell was Fermat thinking when he wrote down x^n + y^n = z^n attached with his very concise proof ?
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  4. #19
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    You should probably want to open a thread to Pre-prints and other original work sub-forum. This is a quite interesting problem (and eventually) with its proof.
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  5. #20
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    Quote Originally Posted by Media_Man View Post
    Attached is a write-up of a final proof (it's three pages long). This problem was fascinatingly complicated. I knew virtually nothing about the Pell equation and continued fractions, so I have had the pleasant opportunity these last few days to understand how they work. Number theory really is enchanting the way a simple question can lead to an arbitrarily complex answer.

    havaliza: Where the hell did this question come from? A book? A class? Your own wanderings?

    After reading and understanding the attached proof, I wonder if anyone wants to look at a generalization involving x(y+n),y(x+n)...
    I can't believe that this problem was this much hard!!
    Thank you for solving that. It was a problem from a High School mathematics contest in Iran. I'm also a high school student. I was looking for a simple understandable proof!!
    Here is the problems' PDF file
    It's the problem in the middle of the page 6.

    P.S. If want to see another problem of this contest, look here!!
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