184 History of the Theory op Numbers. [Chap, vii 
root of x^=l (mod p), if p is a prime >2, he concluded (p. 165) that p is 
a factor of the numbers under the radical signs in the formula for a primitive 
pth root of unity. Cf. Smith^^^ of Ch. VIII. 
Poinsot^^" again treated the same subject. 
J. Ivory^^ stated that a primitive root of a prime p satisfies x^^~^^^^^ — 1, 
but no one of the congruences x'=—l (mod p), <=(p — l)/(2a), where a 
ranges over the odd prime factors of p — 1 ; while a number not a primitive 
root satisfies at least one of the a;' = — 1 . Hence if each a' ^ — 1 and 
^(p-i)/2^ — 1, then a is a primitive root. 
V. A. Lebesgue^^ stated that prior to 1829 he had given in the Bulletin 
du Nord, Moscow, the congruence X = of Cauchy^^ for the integers 
belonging to the exponent n modulo p. 
A. Cauchy^^ proved the existence of primitive roots of a prime p, essen- 
tially as in Gauss' second proof. If p — 1 is divisible by n = a"6V . . . , where 
a, b, c,. . . are distinct primes, he proved that the integers belonging to the 
exponent n modulo p coincide with the roots of 
Y (a^"-l)(a:"^°^-l)(a;"/°--l)... _^ , , , 
^=(x"/"-l)(a:''/^-l)...(a:"/-^-l)...=^ ^^^^ P^' 
The roots of the equation X = are the primitive nth roots of unity. For 
the above divisor n of p — 1, the sum of the Zth powers of the primitive 
roots of a;" = 1 (mod p) is divisible by p if Z is divisible by no one of the 
numbers 
n, n/a, n/h, . . . , n/ab, . . . , n/ahc, .... 
But if several of these are divisors of I, and if we replace n, a, b,. . . by 
<t>{n), 1—a, 1 — 6, . . . in the largest of these divisors in fractional form, we 
get a fraction congruent to the sum of the Ith. powers. In case x'" = l 
(mod p) has m distinct integral roots, the sum of the lib. powers of all the 
roots is congruent modulo p to m or 0, according as I is or is not a multiple 
of m. 
M. A. Stern^^ proved that the product of all the numbers belonging to 
an exponent d is = 1 (mod p) , while their sum is divisible by p if d is divisible 
by a square, but is = ( — 1)" if d is a product of n distinct primes (generaliza- 
tions of Gauss, D. A., arts. 80, 81). If p = 2n+l and a belongs to the expo- 
nent n, the product of two numbers, which do not occur in the period of a, 
occurs in the period of a. To find a primitive root of p when p — 1 = 2ab . . . , 
where a,b,. . . are distinct odd primes, raise any number as 2 to the powers 
(p — l)/a, (p — 1)/6, . . . ; if no one of the residues modulo p is 1, the negative 
of the product of these residues is a primitive root of p; in case one of the 
residues is 1, use 3 or 5 in place of 2. If p = 2g+l and q are odd primes, 2 
or — 2 is a primitive root of p according as p = 8n + 3 or 8n + 7 . If p = 4^^ -f-1 
"''Jour, de I'dcole polytechnique, cah. 18, t. 11, 1820, 345-410. 
"Supplement to Encyclopaedia Britannica, 4, 1824, 698. 
"Jour, de Math., 2, 1837, 258. 
"Exercices de Math., 1829, 231; Oeuvrea, (2), 9, 266, 278-90. 
"Jour, fiir Math., 6, 1830, 147-153. 
