Implementation [of 3/2 system] The evolution for t steps of the system at the top of the page can be computed simply by NestList[If[EvenQ[#], 3#/2, 3(# + 1)/2] &, 1, t]

Complements of recursively enumerable sets are characteristically associated with Π 1 statements of the form ∀ t ϕ [t] —an example being whether a given system never halts. ( Π 1 and Σ 1 statements are such that if they can be shown to be undecidable, then respectively they must be true or false, as discussed on page 1167 .) If a statement in minimal form involves n alternations of ∃ and ∀ it is Σ n + 1 if it starts with ∃ and Π n + 1 if it starts with ∀ . … (Showing that a statement with n ≥ 1 is undecidable does not establish that it is always true or always false.)

Pierre Fermat suggested 2 2 n + 1 as a source for primes and Marin Mersenne 2^Prime[n] - 1 (see page 911 ). … There continued to be slow progress in finding specific large primes; 2 31 - 1 was found prime around 1750 and 2 127 - 1 in 1876. ( 2 2 5 + 1 was found composite in 1732, as have now all 2 2 n + 1 for n ≤ 32 .) … The number of digits in the largest known prime has historically increased roughly exponentially with time over the past two decades, with a prime of over 4 million digits ( 2 13466917 - 1 ) now being known (see page 911 ).

A particularly well-studied example (see page 918 ) is the so-called logistic map x  a x (1 - x) . The base 2 digit sequences obtained with this map starting from x = 1/8 are shown below for various values of a . … The detailed behavior is different for every value of a , but whenever the repetition period is 2 j , it turns out that with any initial condition the leftmost digit always eventually follows a sequence that consists of repetitions of step j in the evolution of the substitution system {1  {1, 0}, 0  {1, 1}} starting either from {0} or {1} .

The pictures below show which sequences of 0's and 1's correspond to complete numbers in these representations. Every vertical column is a possible sequence of 0's and 1's, and the column is shown to terminate when a complete number is obtained. With an infinite random sequence of 0's and 1's, different number representations yield different distributions of sizes of numbers.

Counting of [network] nodes The number of nodes reached by going out to network distance r (with r > 1 ) from any node in the networks on page 477 is (a) 4r - 4 , (b) 3r 2 /2 - 3r/2 + 1 , and (c) First[Select[4r 3 /9 + 2r 2 /3 + {2, 5/3, 5/3} r - {10/9, 1, -4/9}, IntegerQ]] In any trivalent network, the quantity f[r] obtained by adding up the numbers of nodes reached by going distance r from each node must satisfy f[0] = n and f[1] = 3n , where n is the total number of nodes in the network.

The key idea is to represent data of any kind by a symbolic expression of the general form head[arg 1 , arg 2 , …] . ( a + b 2 is thus Plus[a, Power[b, 2]] , {a, b, c} is List[a, b, c] and a = b + 1 is Set[a, Plus[b, 1]] .)

BesselJ[0, x] goes like Sin[x]/ √ x for large x while AiryAi[-x] goes like Sin[x 3/2 ]/x 1/4 . … (For AiryAi[x] the Stokes lines are in directions (-1)^({1, 2, 3}/3) .)

These numbers can also be obtained as the coefficients of x n in the series expansion of x ∂ x Log[ ζ [m, x]] , with the so-called zeta function, which is always a rational function of x , given by ζ [m_, x_] := 1/Det[IdentityMatrix[Length[m]] - m x] and corresponds to the product over all cycles of 1/(1 - x n ) .

Numbers of reversible [cellular automaton] rules For k = 2 , r = 1 , there are 6 reversible rules, as shown on page 436 . … For k = 3 , r = 1 there are 1800 reversible rules, in 172 families. For k = 4 , r = 1 , some of the reversible rules can be constructed from the second-order cellular automata below.