What's the difference between the following two codes? You try Solve[{( m x + y - 7 z)^2 ==
Cos[[Pi]/3]^2*(x^2 + y^2 + z^2)*(m^2 + 1^2 + 7^2), -10 <= m <=
10, 0 <= x <= 10, 0 <= y <= 10, 0 <= z <= 10}, {m, x, y,
z}, Integers]
The number of solutions are more your code.