Primitive recursive functions are defined to deal with non-negative integers and to be set up by combining the basic functions z = 0 & (zero), s = # + 1 & (successor) and p[i_] := Slot[i] & (projection) using the operations of composition and primitive recursion f[0, y___Integer] := g[y] f[x_Integer, y___Integer] := h[f[x - 1, y], x - 1, y] Plus and Times can then for example be defined as plus[0, y_] = y; plus[x_, y_] := s[plus[x - 1, y]] times[0, y_] = 0; times[x_, y_] := plus[times[x - 1, y], y] Most familiar integer mathematical functions also turn out to be primitive recursive—examples being Power , Mod , Binomial , GCD and Prime .

If only one color of element ever appears this is the complete condition for a solution—and for r = 2 solutions exist if Apply[Times, d] < 0 and are then of length at least Apply[Plus[##]/GCD[##]&, Abs[d]] .

Diophantine equations If variables appear only linearly, then it is possible to use ExtendedGCD (see page 944 ) to find all solutions to any system of Diophantine equations—or to show that none exist.