And perhaps if we could only find the appropriate new method it would suddenly be clear that some of what we thought was random radio noise is actually the output of beacons set up by extraterrestrial intelligence.

For while in the past it might have seemed that the only way to generate primes was by using intelligence, we now know that the rather straightforward computations required can actually be carried out by a vast range of different systems—with no apparent need for intelligence.

A sequence of much faster methods have however been developed over the past few decades, one simple example that works for most n being the so-called rho method of John Pollard (compare the quadratic residue sequences discussed below): Module[{f = Mod[# 2 + 1, n] &, a = 2, b = 5, c}, While[(c = GCD[n, a - b])  1, {a, b} = {f[a], f[f[b]]}]; c] Most existing methods depend on facts in number theory that are fairly easy to state, though implementing them for maximum efficiency tends to lead to complex programs. … Nevertheless, to test whether a number is prime ( PrimeQ ) it is known that only a few more than Log[n] steps suffice.

In the first two pictures below, bands with spacing 1/2 Csc[ θ /2] are visible wherever lines cross. … The patterns are exactly repetitive only when Tan[ θ ]  u/v , where u and v are elements of a primitive Pythagorean triple (so that u , v and Sqrt[u 2 + v 2 ] are all integers, and GCD[u, v]  1 ). This occurs when u=r 2 - s 2 , v = 2 r s (see page 945 ), and in this case the minimum displacement that leaves the whole pattern unchanged is {s, r} .

In its typical form, it involves not only explaining natural phenomena by analogy to human behavior but also assuming that they can be influenced as humans might be, say by offerings or worship.

But in general one can make replacements not only for variables in s and t , but also for ones in w .

Mechanical randomness It takes only small imperfections in dice or roulette wheels to get substantially non-random results (see page 971 ).

In the early 1600s Galileo noted that the force of gravity seems to depend only on the mass of an object, and not on any of its other features. … (This principle requires only slightly more than Galileo's idea of the equivalence of gravitational and inertial mass, which has now been verified to the 10 -12 level.) … In the late 1800s Ernst Mach had argued that phenomena like acceleration and rotation could ultimately be defined only relative to matter in the universe.

No doubt there will at first be a tendency to follow the progression of scientific history and to present the ideas of this book only at an advanced stage in the educational process, after teaching many aspects of traditional science. … In many fields, advanced education seems useful only if one intends to pursue those specific fields.

Such models inevitably handle only computable reals (in the sense defined above), and can never do computations beyond those possible in ordinary discrete systems. … It is then usually assumed, however, that the primitive operations performed on these registers are just those of ordinary arithmetic, with the result that only a very limited set of functions (not including for example the exponential function) can be computed in a finite number of steps.