Chap. VII] PRIMITIVE RoOTS, EXPONENTS, INDICES. 197 
J. Perott^^ employed the sum Sk of the ^th powers of 1, 2, . . ., p — 1, and 
gave a new proof that Si=0, . . ., Sp_2=0, Sp_i= —1 (mod p). If m is the 
1. c. m. of the exponents to which 1,2,. . ., p — 1 belong, evidently Sm=p — 1, 
whence m>p — 2. If A belongs to the exponent m, then A, A^, . . ., A"" are 
incongruent, whence mSp — 1- Thus A is a primitive root. 
N. Amici^^ proved that, if j'>2, a number belongs to the exponent 2""^ 
modulo 2" if and only if it is of the form 8/i±3, and called such numbers 
quasi primitive roots of 2\ For a base Sh=^S, numbers of the two forms 
8A:+1 or 8^=*=3, and no others, have indices. The product of two numbers 
having indices has an index which is congruent modulo 2""^ to the sum of 
the indices of the factors. The product of two numbers 6i and 62? neither 
with an index, has an index congruent modulo 2""^ to the sum of the indices 
of —61 and —62- The product of a number with an index by one without 
an index has no index. 
K. Zsigmondy^^ proved by use of abelian groups that, if 8 = qi^\ . .Qr'"', 
m = pi'^K . .ps^s, where Qi,. . ., Qr are distinct primes, and Pi,..., Ps are dis- 
tinct primes, the number of incongruent integers belonging to the exponent 
5 modulo m is 
5i...5,n(l-l/g/0, 
1=1 
where d, is the g. c. d. of 5 and tj=(f>{pp), while li is the number of the 
integers ti,...,ts which contain the factor ql'K 
E. de Jonquieres^^ proved that the product of an even number of primi- 
tive roots of a prime p is never a primitive root, while the product of an 
odd number of them is either a primitive root or belongs to an exponent not 
dividing {p — l)/2. Similar results hold for products of numbers belonging 
to like exponents. Certain of the n integers r, for which f is a given num- 
ber belonging to the exponent e = {p — \)/n, belong to the exponent ne, 
while the others (if any are left) belong to an exponent ke, where k divides n. 
He conjectured that 2 is not a primitive root of a prime p=l, 7, 17 or 23 
(mod 24); 3 not of p=l, 11, 13 or 23 (mod 24); 5 not of p=\, 11, 19, or 
29 (mod 30). These results and analogous ones for 7 and 11 were shown 
by him and T. Pepin^^ to follow from the quadratic reciprocity law and 
Gauss' theorems on the divisors oi x^^A. 
G. Wertheim^°. added to his^* corollaries cases when 6, 10, 11, 13 are 
primitive roots of primes 2^+1, 4^+1; also, 10 is a primitive root of all 
primes 8g+l?^137 for which g- is a prime 10A;+7 or lOyc+9, and of primes 
IGg+l for which g is a prime 10/c+l or lO/c+7. 
Wertheim^^ gave the least primitive root of each prime between 3000 and 
5000 and of certain higher primes. He noted errata in his^^ table to 3000. 
85BuU. des Sc. Math6matiques, 18, I, 1894, 64-66. 
8«Rendiconti Circolo Mat. di Palermo, 8, 1894, 187-201. 
"Monatshefte Math. Phys., 7, 1896, 271-2. 
88Compte8 Rendus Paris, 122, 1896, p. 1451, p. 1513; 124, 1897, p. 334, p. 428. 
8»Comptes Rendus Paris, 123, 1896, pp. 374, 405, 683, 737. 
'"Acta Math., 20, 1896, 143-152. 
"/bid., 153-7; corrections, 22, 1899, 200. 
