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Math Help - Number Theory (1)

  1. #1
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    Number Theory (1)

    This interesting question is taken from an Iranian () competetion (high school level): (Have fun!)

    Suppose x,y,z are positive integers with xy=z^2+1. Prove that there exist integers a,b,c,d such that x=a^2+b^2, \ y=c^2+d^2, and z=ac+bd.
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  2. #2
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    you mean

    (a^2 + b^2 )(c^2 + d^2 ) = (ac + bd )^2 + 1   \implies

     (ad - bc )^2 = 1

     ad - bc = 1 or  bc - ad = 1

    I remember that If  (A,B) = 1 , then there exists integers

     A,B such that  Ax + By = 1 , (x,y) \in \mathbb{N}

    Is it correct if  (a,b) = (a,c) = (b,d) = (c,d) = 1

    there also exists integers  a,b,c,d satisfying

     ad - bc = 1 or  bc - ad = 1 ?


    For exmaple , set  a = 5 , c = 3

     5d - 3b = 1 , i obtain  b = 3 + 5t , d = 2 + 3t

    let t = 0

    therefore

     x = a^2 + b^2 =  25 + 9 = 34

     y = c^2 + d^2 = 9 + 4 = 13

     z = ac + bd = 15 + 6 = 21

     34 \times 13 = 442 = z^2 + 1

    It is also true when  t = 1

     a = 5 , b= 8 , c=3 , d=5

    (5^2 + 8^2 )( 3^2  + 5^2 ) = \left ( (5)(3) + (8)(5) \right)^2 + 1
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  3. #3
    MHF Contributor Bruno J.'s Avatar
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    Oops
    Last edited by Bruno J.; November 22nd 2009 at 10:04 PM.
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  4. #4
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    Quote Originally Posted by Bruno J. View Post

    ...and z=ac+bd follows by simple computation.
    are you sure about that?
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