1911 Encyclopædia Britannica/Number/Exponents, Primitive Roots, Indices

29. Exponents, Primitive Roots, Indices. Let $$p$$ denote an odd prime, and $$x$$ any number prime to p. Among the powers $$x, x^2, \dots x^{p-1}$$ there is certainly one, namely $$x^{p-1}$$, which $$\equiv 1 \pmod{p}$$; let $$x^e$$ be the lowest power of $$x$$ such that $$x^e \equiv 1 \pmod{p}$$. Then $$e$$ is said to be the exponent to which $$x$$ appertains $$\pmod{p}$$: it is always a factor of $$(p-1)$$ and can only be $$1$$ when $$x \equiv 1$$. The residues $$x$$ for which $$e = p-1$$ are said to be primitive roots of $$p$$. They always exist, their number is $$\phi(p-1)$$, and they can be found by a methodical, though tedious, process of exhaustion. If $$g$$ is any one of them, the complete set may be represented by $$g, g^a, g^b, \dots \!\!\And\!\!\!\!\mathrm{c}.$$ where $$a, b, \!\!\And\!\!\!\!\mathrm{c}.$$, are the numbers less than $$(p-1)$$ and prime to it, other than $$1$$. Every number $$x$$ which is prime to $$p$$ is congruent, $$\pmod{p}$$, to $$g^i$$, where $$i$$ is one of the numbers $$1, 2, 3, ... (p-1$$); this number $$i$$ is called the index of $$x$$ to the base $$g$$. Indices are analogous to logarithms: thus
 * $$\operatorname{ind}_g(xy)\equiv\operatorname{ind}_g x+\operatorname{ind}_g y$$. $$\operatorname{ind}_g(x^h)\equiv h\ \operatorname{ind}_g x \pmod{\overline{p-1}}$$

Consequently tables of primitive roots and indices for different primes are of great value for arithmetical purposes. Jacobi's Canon Arithmeticus gives a primitive root, and a table of numbers and indices for all primes less than $$1000$$.

For moduli of the forms $$2p, p^m, 2p^m$$ there is an analogous theory (and also for $$2$$ and $$4$$); but for a composite modulus of other forms there are no primitive roots, and the nearest analogy is the representation of prime residues in the form $$\alpha^x\beta^y\gamma^z \dots$$, where $$\alpha, \beta, \gamma, \dots$$ are selected prime residues, and $$x, y, z, \dots$$ are indices of restricted range. For instance, all residues prime to $$48$$ can be exhibited in the form $$5^x 7^y 13^z$$, where $$x=0, 1, 2, 3$$; $$y = 0, 1$$; $$z=0, 1$$; the total number of distinct residues being $$4.2.2 = 16 =\phi(48)$$, as it should be.