And most methods manage to compress behavior that is repetitive, and at least to some extent behavior that is nested—exactly the two kinds of simple behavior that we have noted many times in this book.

In modern times computer languages have often been thought of as providing precise ways to represent processes that might otherwise be carried out by human thinking.

And indeed, as I have discussed several times in this book, it is in many cases clear that the whole notion of continuity is just an idealization—although one that happens to be almost required if one wants to make use of traditional mathematical methods.

And in fact particularly compared to what I do in this book the vast majority of mathematics practiced today still seems to follow remarkably closely the traditions of arithmetic and geometry that already existed even in Babylonian times.

Typical running times for FactorInteger[n] in Mathematica 4 are shown below for the first 1000 numbers with each of 15 through 30 digits.

From its inception in classical times, through its development in the 1600s to 1800s, number theory was largely separate from other fields of mathematics.

One can always just use n copies of the same symbol to represent an integer n —and indeed this idea seems historically to have arisen independently quite a few times. … The idea of labelling entities in geometrical diagrams by letters existed in Babylonian and Greek times.

The maximum halting times for the first few sizes n are {5, 159, 161, 1021, 5419, 315391, 1978213883, 1978213885, 3018415453261} These occur for inputs {1, 2, 5, 10, 26, 34, 106, 213, 426} and correspond to outputs (each themselves maximal for given n ) 2^{3, 23, 24, 63, 148, 1148, 91148, 91149, 3560523} - 1 Such maxima often seem to occur when the input x has the form (20 4 s - 2)/3 (and so has digits {1, 1, 0, 1, 0, … , 1, 0} ). … A few special cases are: f[4s] = 4s + 3 f[4s + 1] = 2f[2s] + 1 f[2 s - 1] = 2 (10s + 5 + 3 (-1) s )/4 - 1 How the halting times behave for large n is not clear.

Given p = Array[Prime, Length[list], PrimePi[Max[list]] + 1] or any list of integers that are all relatively prime and above Max[list] (the integers in list are assumed positive) CRT[list_, p_] := With[{m = Apply[Times, p]}, Mod[Apply[Plus, MapThread[#1 (m/#2)^EulerPhi[#2] &, {list, p}]], m]] yields a number x such that Mod[x, p]  list .

Instead, as discussed on page 940 , a PDE is essentially just a constraint on the values of a function at different times or different positions.