The formulas we have used so far can be thought of as always consisting of sums of products of variables.

As discussed on page 938 a finitely presented group or semigroup can be viewed as a special case of a multiway system, in which the rules of the multiway system are obtained from relations between strings consisting of products of generators. The word problem then asks if a given product of such generators is equal to the identity element. … Even if a group ultimately has only a finite number of distinct elements, its word problem (with elements specified as products of generators) may still be undecidable.

Rate equations In standard chemical kinetics one assumes that molecules are uniformly distributed in space, so that the rates for particular reactions are proportional to the products of the densities of the molecules that react in them.

For prime k , each cycle (except all 0's) corresponds to a term in the product Factor[x k n - 1 - 1, Modulus  k] .

Many of the pigments used by animals are by-products of metabolism, suggesting that at least at first pigmentation patterns were probably often incidental to the operation of the animal.

And in the 1990s Ivan Korec and others showed that it could be done just with Mod[Binomial[a + b, a], k] with k = 6 or any product of primes—and that it could not be done with k a prime or prime power.

f[n_] := n f[n - 1]; f[1] = 1 f[n_] := Product[i, {i, n}] f[n_] := Module[{t = 1}, Do[t = t i, {i, n}]; t] f[n_] := Module[{t = 1, i}, For[i = 1, i ≤ n, i++, t ⋆ = i]; t] f[n_] := Apply[Times, Range[n]] f[n_] := Fold[Times, 1, Range[n]] f[n_] := If[n  1, 1, n f[n - 1]] f[n_] := Fold[#2[#1] &, 1, Array[Function[t, # t] &, n]] f = If[#1  1, 1, #1 #0[#1 - 1]] &

The smallest product of these numbers is 24 (compare note below), and the rule he gave in this case is: Note that these results concern Turing machines which can halt (see page 1137 ); the Turing machines that I consider do not typically have this feature.

Gaussian distributions typically arise when measurements involve sums of random quantities; other common distributions are obtained from products or other simple combinations of random quantities, or from the results of simple processes based on random quantities.

The main point, I believe, is that in both the systems it studies and the questions it asks mathematics is much more a product of its history than is usually realized.