½ = 1/x² + 1/y² +...+ 1/z²

All variables must be different, positive integers, and there must be a finite number of terms.

The answer is that there is no solution.

Clearly one of the integers cannot be 1 or it would be too large. The largest possible sum is from 2 to infinity. Since it's a p-series with p>1, the series converges. To show that it's not possible, we need to show that the largest possible sum is less than 1/2.

Consider the function f(x)=1/x^2. The integral from 2 to infinity is 1/2. Since f(x)>0, this happens to be the area as well. The series that we are looking for happens to be the right-hand Reimann sum for this integral with a unit differential. Geometrically, this is the sum of the areas of the rectangles, where the nth rectangle has a base of 1 and a height of 1/(n+1)^2.

Because this is a monotonically decreasing function, the right hand Reimann sum will be less than the actual area. Conversely, if it were a left-hand sum, it'd be larger than the actual area. Hence, the largest possible sum is less than the integral, which is 1/2. Therefore, there is no possible solution to the equation given the restrictions.