For multiplication rules, there are normally carries (handled by FromDigits ), but for power cellular automata, these have only limited range, so that g = Mod[#, k α ] & can be used.

In general one can imagine characterizing the power of any axiom system by giving a transfinite number κ which specifies the first function  [ κ ] (see note above ) whose termination cannot be proved in that axiom system (or similarly how rapidly the first example of y must grow with x to prevent ∃ y p[x, y] from being provable).

In the late 1970s it was noted that by evaluating PowerMod[a, n - 1, n]  1 for several random integers a one can with high probability quickly deduce PrimeQ[n] .

Most of the programs require only the language component of Mathematica—and not its mathematical knowledge base—and so should run in all software systems powered by Mathematica, in which language capabilities are enabled.

From this representation of Power the universal equation can be converted to a purely polynomial equation with 2154 variables—which when expanded has 1683150 terms, total degree 16 (average per term 6.8), maximum coefficient 17827424 and LeafCount 16540206.

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 .

To find predictions from this theory a so-called perturbation expansion was made, with successive terms representing progressively more interactions, and each having a higher power of the so-called coupling constant α ≃ 1/137 .