Then in 1847 Ernst Kummer used ideas of factoring with algebraic integers to prove it for all n < 37 .

(The groups can be written as products of cyclic ones whose orders correspond to the possible factors of n .)

(For non-prime k , the patterns are obtained by superimposing the patterns corresponding to the factors of k .)

If k is not prime the pattern is no longer strictly invariant with respect to keeping only every k th row and column—but is in effect still a superposition of patterns with this property for factors of k .

And as it turns out, the repetition period is again related to the factors of the number of possible positions for the dot—and tends to be maximal in those cases where this number is prime.

But in the end history seems to be the only real determining factor.

Its computation is known in general to be equivalent in difficulty to factoring n (see page 1090 ).

Encoding only differences between successive samples leads to perhaps a factor of 2 compression.

The evolution of the arithmetic system is given by ASEvolveList[{n_, rules_}, init_, t_] := NestList[(Mod[#, n] /. rules)[#] &, init, t] Given a value m obtained in the evolution of the arithmetic system, the state of the register machine to which it corresponds is {Mod[m, p] + 1, Map[Last, FactorInteger[ Product[Prime[i], {i, nr}] Quotient[m, p]]] - 1} Note that it is possible to have each successive step involve only multiplication, with no addition, at the cost of using considerably larger numbers overall.

Charles Spearman suggested in 1904 that there might be a general intelligence factor (usually called g ) associated with all intellectual tasks.