SOLUTION: Hi, I don't know how to approack this question other than try guess and check... Sorry, but here's the question: Prove that there are no positive integers x and y such that: 1/x^

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: Hi, I don't know how to approack this question other than try guess and check... Sorry, but here's the question: Prove that there are no positive integers x and y such that: 1/x^      Log On


   

Question 1046170: Hi, I don't know how to approack this question other than try guess and check... Sorry, but here's the question:
Prove that there are no positive integers x and y such that:
1/x^2 + 1/xy + 1/y^2 = 1
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Suppose that THERE ARE positive integers x and y such that
1%2Fx%5E2+%2B+1%2F%28xy%29+%2B+1%2Fy%5E2+=+1.
===> 1%2Fx%5E2+%2B+2%2F%28xy%29+%2B+1%2Fy%5E2+=+1%2F%28xy%29+%2B1+=+%28xy%2B1%29%2F%28xy%29.
<===> %281%2Fx+%2B+1%2Fy%29%5E2+=+%28xy%2B1%29%2F%28xy%29.
===> %28%28+x%2By%29%2F%28xy%29%29%5E2+=+%28xy%2B1%29%2F%28xy%29.
===> %28x%2By%29%5E2%2F%28xy%29%5E2+=+%28xy%2B1%29%2F%28xy%29.
<===> %28xy%29%28xy%2B1%29+=+%28x%2By%29%5E2
<===> %28xy%29%5E2+%2B+%28xy%29+-+%28x%2By%29%5E2+=+0

===> xy+=+%28-1+%2B-+sqrt%28+1%2B4%28x%2By%29%5E2+%29%29%2F2+
===> xy+=+%28-1+%2B+sqrt%28+1%2B4%28x%2By%29%5E2+%29%29%2F2+ , since xy is a positive integer.

Because of the denominator 2, and the -1 in the numerator, we are forced to say that
1%2B4%28x%2By%29%5E2+ is an ODD PERFECT SQUARE.
===>+1%2B4%28x%2By%29%5E2++=+%282M%2B1%29%5E2 for some positive integer M.
<===> 1%2B4%28x%2By%29%5E2++=+4M%5E2+%2B4M+%2B1
===> %28x%2By%29%5E2+=+M%5E2%2BM+=+M%28M%2B1%29,
which says that a perfect square (%28x%2By%29%5E2) is the product of two consecutive positive integers ( M(M+1) ). But this is impossible.
Hence, a contradiction.
Therefore there CANNOT be two positive integers such that 1%2Fx%5E2+%2B+1%2F%28xy%29+%2B+1%2Fy%5E2+=+1.