One can compute the result of 9 steps of evolution as 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 , but a better scheme is to use partial results and compute successively 1 + 1; 2 + 2; 1 + 4; 5 + 5 —which is what the repeated squaring method above does when h = Plus , r = s = 1 . … But to be able to do this some of the arguments given to h inevitably need to be larger.

But to do it approximately is considerably easier.

Almost certainly this does indeed happen in at least some critical situations.

It is even conceivable that a Diophantine equation with 2 variables could be universal: with one variable essentially being used to represent the program and input, and the other the execution history of the program—with no finite solution existing if the program does not halt.

For while they make it easy at a formal level to check that certain statements are true, they do little at a more conceptual level to illuminate why this might be so.

With a rule given in this form, a single step in the evolution of the Turing machine can be implemented with the function TMStep[rule_List, {s_, a_List, n_}] /; (1 ≤ n ≤ Length[a]) := Apply[{#1, ReplacePart[a, #2, n], n + #3}&, Replace[{s, a 〚 n 〛 }, rule]] The evolution for many steps can then be obtained using TMEvolveList[rule_, init_List, t_Integer] := NestList[TMStep[rule, #]&, init, t] An alternative approach is to represent the complete state of the Turing machine by MapAt[{s, #}&, list, n] , and then to use TMStep[rule_, c_] := Replace[c, {a___, x_, h_List, y_, b___}  Apply[{{a, x, #2, {#1, y}, b}, {a, {#1, x}, #2, y, b}} 〚 #3 〛 &, h /. rule]] The result of t steps of evolution from a blank tape can also be obtained from (see also page 1143 ) s = 1; a[_] = 0; n = 0; Do[{s, a[n], d} = {s, a[n]} /. rule; n += d, {t}]

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 .

But for the bottom example, the pictures below show that convergence is not so rapid, and indeed, as is typical in working with PDEs, despite having used large amounts of computer time I do not know whether the details of the picture in the main text are really correct.

The set of values of c for which it does corresponds exactly to the boundary of the Mandelbrot set.

And although statistical significance is reduced by considering only discrete features, some evidence has emerged that different species do indeed have shapes related by changes in fairly small numbers of geometrical parameters.