For while in the past it might have seemed that the only way to generate primes was by using intelligence, we now know that the rather straightforward computations required can actually be carried out by a vast range of different systems—with no apparent need for intelligence.

As I discussed earlier, the more direct the representation the more easily an ordinary physical process can be expected to generate it, and the less there will be any indication of intelligence—just as, for example, something like a photograph can be produced essentially just by projecting light, while a diagram or a painting requires more.

For while this is undoubtedly very common say in cellular automata, the most immediate suggestions of it are in class 4 systems like rule 110 that in effect happen to do their computations in a way that looks at least somewhat similar to the way we as humans are used to doing them.

A sequence of much faster methods have however been developed over the past few decades, one simple example that works for most n being the so-called rho method of John Pollard (compare the quadratic residue sequences discussed below): Module[{f = Mod[# 2 + 1, n] &, a = 2, b = 5, c}, While[(c = GCD[n, a - b])  1, {a, b} = {f[a], f[f[b]]}]; c] Most existing methods depend on facts in number theory that are fairly easy to state, though implementing them for maximum efficiency tends to lead to complex programs.

Comparison to multiway systems Operator systems are normally based on equations, while multiway systems are based on one-way transformations.

If no self connections are allowed then these numbers become {1, 2, 6, 20, 91} , while if neither self nor multiple connections are allowed (yielding what are often referred to as cubic or 3-regular graphs), the numbers become {0, 1, 2, 5, 19, 85, 509, 4060, 41301, 510489} , or asymptotically (6 n)!

And while methods for their cryptanalysis were developed in the 1400s, such systems continued to see occasional serious use until the early 1900s. … Initially several different problems were considered, but after a while the only ones to survive were those such as the RSA system discussed below based essentially on the problem of factoring integers.

Mathematical notation While it is usually recognized that ordinary human languages depend greatly on history and context, it is sometimes believed that mathematical notation is somehow more universal. … And while traditional mathematical notation suffers from some inconsistencies and ambiguities, it was possible in developing Mathematica StandardForm to set up something very close that can be interpreted uniquely in all cases.

And while this problem has resisted a fair number of attempts at solution, it is not known to be NP-complete (and indeed its ability to be solved in polynomial time on a formal quantum computer may suggest that it is not).

Based on this LE[list_] := Module[{n = Length[list], i = Max[MapIndexed[ #1 - #2 &, PrimePi[list]]] + 1}, CRT[PadRight[ list, n + i], Join[Array[Prime[i + #] &, n], Array[Prime, i]]]] will yield a number x that can be decoded into a list of length n using essentially the so-called Gödel β function Mod[x, Prime[Rest[NestList[NestWhile[# + 1 &, # + 1, Mod[x, Prime[#]]  0 &] &, 0, n]]]]