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Find the Integers - Answer

[X, Y, Z] = [17, 8, -8]

If we take the sums and differences of the equations involving X and Y, we find that 2X is the sum of two squares in two different ways, 2Z is the difference of two squares in two different ways, and 2Y is both the sum of two squares and the difference of two squares.

Then we know that the squared integers in the equations for 2X, 2Y, and 2Z must be either both even or both odd.

Since we are looking for the minimum solution for X + Y + Z, take cases in which the integers that are squared are small, perhaps 0 through 10. Working through those values yields the answer given above.