Chap. VII] PRIMITIVE RoOTS, EXPONENTS, INDICES. 203 
To this congruence he appHed Hurwitz's^^ method (Ch. VIII) of finding the 
number of roots and concluded that there are p — l—(f>(p — l) roots. 
Hence there exist 0(p — 1) primitive roots of p. 
A. Cunningham and H. J. WoodalP^^ continued to p< 100000 the table 
of Cunningham"" of the maximum residue indices j^ of 2 modulo p. 
C. Posse^^^ reproduced Korkine's"^ and his own"^ tables and explained 
their use in the solution of binomial congruences. 
C. Krediet^^o treated x*'=l (mod n) of Lucas/^" Ch. Ill, and called x 
a primitive root if it belongs to the exponent cp. The powers of such a 
root are placed at equal intervals on a circle for various n's. 
G. A. Miller^^^ proved by use of group theory that, if m is arbitrary, 
the sum of those integers < m and prime to m which belong to an exponent 
divisible by 4 is = (mod m) , and the sum of those belonging to the expo- 
nent 2 is = — 1 (mod m), and proved the corresponding theorem by Stern^^ 
for a prime modulus. 
A. Cunningham^^^ tabulated the number of primes p<10^ for which 
y belongs to the same exponent modulo p, for y = 2, 3, 5, 6, 7, 10, 11, 12; 
and the number of primes p in each 10000 to 10^ for which y (2/ = 2 or 10) 
belongs to the same exponent modulo p. Also, for the same ranges on 
p and y, the number of primes p for which y''^ 1 (mod p) is solvable, where 
A; is a given divisor of p — 1 . 
A. Cunningham^^^ stated that he had finished the manuscript of a table 
of Haupt-exponents to bases 3, 5, 6, 7, 11, 12 for all prime powers < 15000; 
also canons giving at sight the residues of z" modulo p'''< 10000 for z = 2, 
r^l00;2 = 3, 5, 7, 10, 11, r^30. 
J. Barinaga^^^ considered a number a belonging to the exponent g 
modulo p, a prime. If a is not divisible by g, the sum of the ath powers 
of the numbers forming the period of a modulo p is divisible by p. The 
sum of their products n at a time is congruent to zero modulo p ii n<g, 
but to =^"1 ii n = g, according as g is even or odd. 
A. Cunningham^^^ listed errata in his Binary Canon^^ and Jacobi's Canon. ^^ 
G. A. Miller^^^ employed the group formed by the integers <m and 
prime to m, combined by multiplication modulo m, to show that, if a 
number is = ± 1 (mod 2"^), but not modulo 2^+\ where l<7</3, it belongs 
to the exponent 2^~^ modulo 2^. Also, if p is an odd prime, and A^= 1 
(mod p), N belongs to the exponent p^~^ modulo p^ if and only if A^" — 1 is 
divisible by p^, but not by p^+\ where /3>7^ 1. 
»8Quar. Jour. Math., 42, 1911, 241-250; 44, 1913, 41-48, 237-242; 45, 1914, 114-125. 
i^Acta Math., 35, 1912, 193-231, 233-252. 
""Wiskundig Tijdskrift, Haarlem, 8, 1912, 177-188; 9, 1912, 14-38; 10, 1913, 40-46, 87-97. 
(Dutch.) 
"lAmer. Math. Monthly, 19, 1912, 41-6. 
"2Proc. London Math. Soc, (2), 13, 1914, 258-272. 
"'Messenger Math., 45, 1915, 69. Cf. Cunningham."" 
"<Annaes Sc. Acad. Polyt. do Porto, 10, 1915, 74-6. 
"^Messenger Math., 46, 1916, 57-9, 67-8. 
"s/Wd., 101-3. 
