In a problem like satisfiability, however, difficult instances tend to occur only on the boundary between cases where the density of black or white squares implies that there is usually satisfaction or usually not satisfaction.

(Note that from the axioms of ring theory one can only expect to prove results that hold for any ring; to get most results in number theory, for example, one needs to use the axioms of arithmetic, which are intended to be specific to ordinary integers.)

(The first two of these were identified around the end of the 1800s; the last with clarity only in the 1960s.)

The basic idea is that to each cell is added an arrow, and any pair of cells is updated only when their arrows point at each other.

In many fields outside of statistics, however, the idea persisted even to the 1990s that block frequencies (or flat frequency spectra) were somehow the only ultimate tests for randomness. … To disallow procedures say specially set up to pick out all the infinite number of 1's in a sequence Alonzo Church in 1940 suggested that only procedures corresponding to finite computations be considered. … Starting in the late 1940s the development of information theory began to suggest connections between randomness and inability to compress data, but emphasis on p Log[p] measures of information content (see page 1071 ) reinforced the idea that block frequencies are the only real criterion for randomness.

Pierre Fermat suggested 2 2 n + 1 as a source for primes and Marin Mersenne 2^Prime[n] - 1 (see page 911 ). In 1752 Christian Goldbach showed that no ordinary polynomial could generate only primes, though as pointed out by Leonhard Euler n 2 - n + 41 does so for n < 40 . … 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 .)

Kurt Gödel ): Mathematics involves trying to discover the properties of a world of ideal mathematical forms, of which we in effect perceive only shadows. • Logicism (e.g.

One can have a rule be applied only once using Module[{i = 1}, expr /. lhs  rhs /; i++  1] Many symbolic systems (including the one on page 103 ) have the so-called Church–Rosser property (see page 1036 ) which implies that if a fixed point is reached in the evolution of the system, this fixed point will be the same regardless of the order in which rules are applied.

As indicated in the main text, networks with more than three connections at each node can be emulated by combining nodes into groups, and looking only at the connections between groups.

Deism emphasizes that God can govern the universe only according to natural laws—but whether or not this involves thinking is unclear.