In the first case shown, the total number of elements obtained doubles at every step; in the second case, it follows a Fibonacci sequence, and increases by a factor of roughly (1+Sqrt[5])/2 ≃ 1.618 at every step.

Diophantine equations Any algebraic equation—such as x 3 + x + 1  0 —can readily be solved if one allows the variables to have any numerical value. … But the Pell equation x 2  a y 2 + 1 (already studied in antiquity) has infinitely many solutions whenever a is positive and not a perfect square. … (The equation (2x + 1) y  z also for example has solutions only when z is not of the form 2 j .)

String theory The sequences of symbols I call strings here have absolutely no direct connection to the continuous deformable 1D objects known as strings in string theory.

Out of all possible Boolean functions the number that require BDDs of sizes 1, 2, ... is for n = 2 : {1, 0, 6, 9} and for n = 3 : {1, 0, 0, 27, 36, 132, 60} ; the absolute maximum grows roughly like 2 n .

Shot noise arises from statistical fluctuations in the flow of charge carriers: if a single bit of data is represented by 10,000 electrons, the magnitude of the fluctuations will typically be about 1%. … Flicker (1/f) noise. Almost all electronic devices also exhibit a third kind of noise, whose main characteristic is that its spectrum is not flat, but instead goes roughly like 1/f over a wide range of frequencies.

Statements in Peano arithmetic Examples include: • √ 2 is irrational: ¬ ∃ a ( ∃ b (b ≠ 0 ∧ a × a  ( Δ Δ 0) × (b × b))) • There are infinitely many primes of the form n 2 + 1 : ¬ ∃ n ( ∀ c ( ∃ a ( ∃ b (n + c) × (n + c) + Δ 0  ( Δ Δ a) × ( Δ Δ b)))) • Every even number (greater than 2) is the sum of two primes (Goldbach's Conjecture; see page 135 ): ∀ a ( ∃ b ( ∃ c (( Δ Δ 0) × ( Δ Δ a)  b + c ∧ ∀ d ( ∀ e ( ∀ f ((f  ( Δ Δ d) × ( Δ Δ e) ∨ f  Δ 0) ⇒ (f ≠ b ∧ f ≠ c))))))) The last two statements have never been proved true or false, and remain unsolved problems of number theory. The picture shows spacings between n for which n 2 + 1 is prime.

Instead it is common for a leading digit s in base b to occur with frequency Log[b, (s + 1)/s] (so that in base 10 1's occur 30% of the time and 9's 4.5%).

Symmetric 5-neighbor [2D cellular automaton] rules Among the 32 possible 5-cell neighborhoods shown for example on page 941 there are 12 classes related by symmetries, given by s = {{1}, {2, 3, 9, 17}, {4, 10, 19, 25}, {5}, {6, 7, 13, 21}, {8, 14, 23, 29}, {11, 18}, {12, 20, 26, 27}, {15, 22}, {16, 24, 30, 31}, {28}, {32}} Completely symmetric 5-neighbor rules can be numbered from 0 to 4095, with each digit specifying the new color of the cell for each of these symmetry classes of neighborhoods. Such rule numbers can be converted to general form using FromDigits[Map[Last, Sort[Flatten[Map[Thread, Thread[{s, IntegerDigits[n, 2, 12]}]], 1]]], 2]

Reversible mobile automata can for instance be constructed using Table[(IntegerDigits[i, 2, 3]  If[First[#]  0, {#, -1}, {Reverse[#], 1}]&)[IntegerDigits[perm 〚 i 〛 , 2, 3]], {i, 8}] where perm is an element of Permutations[Range[8]] .

Binomial distribution If black squares appear independently with probability p then the probability that m squares out of n are black is Binomial[n, m] p m (1 - p) n - m .