CHAPTER VII. 
PRIMITIVE ROOTS. BINOMIAL CONGRUENCES. 
Primitive Roots, Exponents, Indices. 
J. H. Lambert^ stated without proof that there exists a primitive root 
g of any given prime p, so that g^ — \ is divisible by p for e = p — 1, but not 
for 0<e<p — 1. 
L. Euler^ gave a proof which is defective. He introduced the term 
primitive root and proved (art. 28) that at most n integers a;<p make 
x" — 1 divisible by p, the proof applying equally well to any polynomial 
of degree n with integral coefficients. He stated (art. 29) that, for n<p, 
x" — 1 has all n solutions "real" if and only if n is a divisor of p — 1; in par- 
ticular, x^~^ — l has p — 1 solutions (referring to arts. 22, 23, where he 
repeated his earlier proof of Fermat's theorem). Very likely Euler had 
in mind the algebraic identity a;^"^ — l = (x" — 1)Q, from which he was in a 
position to conclude that Q has at most n— p+1 solutions, and hence x" — 1 
exactly n. By an incomplete induction (arts. 32-34), he inferred that there 
are exactly </)(n) integers x<p for which x" — 1 is divisible by p, but x^ — \ 
not divisible by p for 0<Z<n, n being a divisor of p — 1 (as the context 
indicates). In particular, there exist <f>{p — l) primitive roots of p (art. 46). 
He listed all the primitive roots of each prime ^ 37. 
J. L. Lagrange^ proved that, if p is an odd prime and 
a:P-i-l=Z^+pF, 
where X, ^, F are polynomials in x with integral coefficients, and if x"" and x" 
are the highest powers of a; in X and ^ with coefficients not divisible by p, 
there are m integral values, numerically <p/2, of x which make X a mul- 
tiple of p, and fi values making ^ a multiple of p. For, by Fermat's theorem, 
the left member is a multiple of p for a; = ± 1, ± 2, . . . , =*= (p — 1)/2, while at 
most m of these values make X a multiple of p and at most fx make ^ a 
multiple of p. 
L. Euler^ stated that he knew no rule for finding a primitive root and 
gave a table of all the primitive roots of each prime ^41. 
Euler^ investigated the least exponent x (when it exists) for which 
fa'+g is divisible by N. Find X such that —g^\N is a multiple, say aV, 
of a. Then fa'-^-r is divisible by N. Set r=FX'iV = a^s, ^^1. Then 
j'gx-a-ff_^ is divisible by N; etc. If the problem is possible, we finally get 
/ as the residue of /a"^""" • • • "^, whence x = a+. . . +f . For example, to find 
the least x for which 2"" — 1 is divisible by iV = 23, we have 
1+23 = 2^3, 3-23= -2^5, -5-23= -2^7, -7-^23 = 2^, 
whence a^ = 3+2+2+4 = ll. 
^Nova Acta Eruditomm, Leipzig, 1769, p. 127. 
''Novi Comm. Acad. Petrop., 18, 1773, 85; Comm. Arith., 1, 516-537. 
^Nouv. M6m. Ac. Roy. Berlin, ann^e 1775 (1777), p. 339; Oeuvres 3, 777. 
<0pu8c. Anal., 1, 1783 (1772),. 121; Comm. Arith., 1, 506. 
^Opusc. Anal., 1, 1783 (1773), 242; Comm. .\i-ith., 2, p. 1; Opera postuma, I, 172-4. 
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