If the values of the cells in each block are labelled p , q and r , then rule 110 can be written as Mod[(1+p) q r + q + r, 2] or And[Not[And[p, q, r]], Or[q, r]].

So, for example, the smallest solution to x 2  61 y 2 +1 is x  1766319049 , y  226153980 , while the smallest solution to x 3 +y 3  z 3 +2 is x  1214928 , y  3480205 , z  3528875 .

The first section of Chapter 1 provides a basic—though compressed—overview of some of key ideas.

Then in 1967 when the first pulsar was discovered it was briefly thought that perhaps its precise 1.33730113-second repetition rate might be of intelligent origin. … Most use the same basic scheme: to look for signals that show a narrow band of frequencies—say only 1 Hz wide—perhaps changing in time.

Boole identified 1 with True and 0 with False , then noted that theorems in logic could be stated as equations in which Or is roughly Plus and And is Times —and that such equations can be manipulated by algebraic means. … In 1931 Mordechaj Wajsberg found the slightly simpler {(a ⊼ (b ⊼ c)) ⊼ (((d ⊼ c) ⊼ ((a ⊼ d) ⊼ (a ⊼ d))) ⊼ (a ⊼ (a ⊼ b)))} Such an axiom system can be converted to an equational one using Module[{a}, With[{t = a ⊼ (a ⊼ a), i = #1 ⊼ (#2 ⊼ #2) &}, Join[Thread[axioms  t], {i[t ⊼ (b ⊼ c), c]  t, i[t, b]  b, i[i[a, b], b]  i[i[b, a], a]}]]] but then involves 4 axioms.

And in d dimensions it is common for it to take, say, t d + 1 steps for one system to emulate t steps of evolution of another.

These differ in frequency by successive factors of roughly 2 1/12 —with different temperament schemes using different rational approximations to powers of this quantity.

With two possible forms of behavior h[i] = 0 or 1 for initial condition i , an example of such a number is Sum[2 -i h[i], {i, ∞ }] .

For k > 1 this yields a nested pattern, analogous to those shown on page 871 .

For successive numbers of elements the total numbers of these are 1, 2, 5, 16, 63, 318, 2045, 16999, ...