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Number Theory

  1. Oct 23, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove that they are no integers a,b,n>1 such that (a^n - b^n) | (a^n + b^n).

    2. Relevant equations



    3. The attempt at a solution
    Do I solve this by contradiction? If so, how do I start it?
     
  2. jcsd
  3. Oct 23, 2009 #2
    I think it is by contradiction (i suppose you could show the gcd of [itex] (a^n - b^n , a^n + b^n ) [/itex] is 1 or 2 ) viz,
    let d be the gcd clearly then , d must divide the sum (and the difference) of the two , [tex] d | a^n + b^n + a^n - b^n [/tex]
    [tex] d | 2a^n [/tex]
    which implies, [itex] d|2, [/itex] [itex] d|a^n [/itex] this last result shows d is either 1 or 2 , thus if the gcd of the two is 2 or 1, no integers a,b, c >1 can exist to satisfy the requirement (there are no numbers a,b, c> 1 that can divide in that manner) though im only into number theory as a hobby, i might not be quite on the mark, ;-) ,
    good luck
     
    Last edited: Oct 23, 2009
  4. Oct 23, 2009 #3

    I consulted with my study partners and we agree that what you have is correct. But we didn't get exactly what you got so we had to make some corrections. thanks for your help.
     
  5. Nov 26, 2009 #4
    (a^n - b^n) | (a^n + b^n)

    what does that mean?? does the bar symbol mean "given"
     
  6. Nov 26, 2009 #5

    Dick

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    The bar symbol means "divides" in the sense of integer divisibility.
     
    Last edited: Nov 26, 2009
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