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Math Help - prove

  1. #1
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    prove

    prove that if a+ b/a - 1/b is an integer then it is a perfect square for integers a, b.
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  2. #2
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    Quote Originally Posted by nh149 View Post
    prove that if a+ b/a - 1/b is an integer then it is a perfect square for integers a, b.
    By setting a=-1, b=1 you can see the statement does not hold.

    But let's try to fix it: "if a+ b/a - 1/b is an integer then it is a perfect square for positive integers a, b."


    So let a+\frac{b}{a}-\frac{1}{b} = a+\frac{b^2-a}{ab} be an integer for positive integers a,b. Then b^2-a=kab for some integer k. We get b^2=a(kb+1).
    We see this implies k \ge 0.
    If k=0 we immediately see that our statement holds.
    We'll finish the proof by showing that we cannot have k>0. If it is, then b^2=a(kb+1) implies a<b. After dividing b^2=a(kb+1) by b we get b=ak+\frac{a}{b}. This means that b divides a, so we cannot have a<b, which is a contradiction.
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