Two sine functions Sin[a x] + Sin[b x] can be rewritten as 2 Sin[1/2(a + b) x] Cos[1/2(a - b) x] (using TrigFactor ), implying that the function has two families of equally spaced zeros: 2 π n/(a + b) and 2 π (n + 1/2)/(b - a) .

[Computing] square roots A standard way to compute √ n is Newton's method (actually used already in 2000 BC by the Babylonians), in which one takes an estimate of the value x and then successively applies the rule x  1/2 (x + n/x) . … Another approach to computing square roots is based on the fact that the ratio of successive terms in for example the sequence f[i] = 2 f[i - 1] + f[i - 2] with f[1] = f[2] = 1 tends to 1 + √ 2 . … Note that the method works not only for integers, but for any rational number n for which 1 ≤ n < 4 .

It was shown by Euclid in 300 BC that 2 n - 1 (2 n - 1) is a perfect number whenever 2 n - 1 is prime. … The values of n for the known Mersenne primes 2 n - 1 are shown below. … Various generalizations of perfect numbers have been considered, requiring for example IntegerQ[DivisorSigma[1, n]/n] (pluperfect) or Abs[DivisorSigma[1, n] - 2n] < r (quasiperfect).

Thus the patterns on page 189 can be formed from t -digit integers in base  - 1 containing only digits 0 and 1, as given by Table[FromDigits[IntegerDigits[s, 2, t],  - 1], {s, 0, 2 t -1}] In the particular case of base  - q with digits 0 through q 2 , it turns out that for sufficiently large t any complex integer can be represented, and will therefore be part of the pattern.

The Sequence of Primes In the sequence of all possible numbers 1, 2, 3, 4, 5, 6, 7, 8, ... most are divisible by others—so that for example 6 is divisible by 2 and 3. … And so for example 5 and 7 are not divisible by any other numbers (except trivially by 1). … One starts on the top line with all numbers between 1 and 100.

by considering digit sequences in which each digit can again have only a discrete set of possible values, typically just 0 and 1. … The picture on the facing page shows what happens with a slightly more complicated rule in which the average gray level is multiplied by 3/2 , and then only the fractional part is kept if the result of this is greater than 1. A continuous cellular automaton in which each cell can have any level of gray between white (0) and black (1).

Nested digit sequences The number obtained from the substitution system {1  {1, 0}, 0  {0, 1}} is approximately 0.587545966 in base 10. … From the result on page 890 , the number whose digits are obtained from {1  {1, 0}, 0  {1}} is given by Sum[2^(-Floor[n GoldenRatio]), {n, ∞ }] .

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)! … If one requires the networks to be planar the numbers are {0, 1, 1, 3, 9, 32, 133, 681, 3893, 24809, 169206} . If one looks at subnetworks with dangling connections, the number of these up to size 10 is {2, 5, 7, 22, 43, 141, 373, 1270, 4053, 14671} , or {1, 1, 2, 6, 10, 29, 64, 194, 531, 1733} if no self or multiple connections are allowed (see also page 1039 ).

(m) The pattern can be generated by a 2D substitution system with rule {1 -> {{0, 0}, {0, 1}}, 0 -> {{1, 1}, {1, 0}}} (see page 583 ).

If k and n have no factors in common, there will be a t for which Mod[k t , n]  1 , so that the dot returns to position 1. … (This value is related to the repetition period for the digit sequence of 1/n in base k , as discussed on page 912 ). When GCD[k, n]  1 the dot can never visit position 0.