[Universality in] equations

For any purely algebraic equation involving real numbers it is possible to find a bound on the size of any isolated solutions it has, and then to home in on their actual values. But as discussed on page 786, nothing similar is true for equations involving only integers, and in this case finding solutions can in effect require following the evolution of a system like a cellular automaton for infinitely many steps. If one allows trigonometric functions, any equation for integers can be converted to one for real numbers; for example x^^{2}+y^^{2}==z^^{2} for integers is equivalent to Sin[Pi x]^^{2} + Sin[Pi y]^^{2} +Sin[Pi z]^^{2} + (x^^{2}+y^^{2}-z^^{2})^^{2}==0 for real numbers.