A subset won’t have more solutions than its containing set; but it may have fewer. Many problems are easier to solve in the reals (due to being complete), and

A subset won’t have more solutions than its containing set; but it may have fewer. Many problems are easier to solve in the reals (due to being complete), and you can then restrict that solution to your (sub)set of interest — in this case, the integers. You see the same thing with Pythagorean triples being simpler to solve by doing the math over the complex numbers and then restricting your answers.