WLOG, x, y and z can all be positive with x<=y<=z, as they can be permuted and negated in any combination without changing the value.
Since x is the smallest integer, 1/x^2 is the largest term. It can't be 1 as that would already exceed 2/3. But if x is 3 or larger, then 1/x^2 would be 1/9 or smaller, and since this is the largest term, the total couldn't be larger than 3/9 = 1/3.
So the smallest integer, which we have assigned to x, must be 2.
Then 1/y^2 + 1/z^2 must equal 2/3 - 1/4 = 5/12.
Again, since y <= z, y can't be 3 or more as 1/9 + 1/9 would fail to be as high as 5/12. So y could only be 2.
But then the third term, 1/z^2, would have to be 2/3 - 1/4 - 1/4 = 1/6, which is not the reciprocal of a perfect square.
Therefore, there can be no set of integers that satisfies the equation.
Posted by Charlie
on 2012-12-04 14:36:19