Chap. VII] PRIMITIVE RoOTS, EXPONENTS, INDICES. 199 
Fd{x) of a;'*— 1 (dividing the last but not x'— 1 iorKd). To every divisor 
d of p — 1 belong exactly 4>{d) numbers which are the roots of i^d(^)=0 
(modp). 
P. G. Foglini^^ gave an exposition of known results on primitive roots, 
indices, linear congruences, etc. In applying (p. 322) Poinsot's^ method of 
finding the primitive roots of a prime p to the case p = 13, it suffices to exclude 
the residues of the cubes of the numbers which remain after excluding the 
residues of squares; for, if a; is a residue of a square, (x^)®=l and x^ is the 
residue of a square. 
R. W. D. Christie^"" noted that, if 7 is a primitive root of a prime 
p = 4A; — l,the remaining primitive roots are congruent to p — 7 (n = 1, 2, . . . ) 
A. Cunningham^°^ noted that 3, 5, 6, 7, 10 and 12 are primitive roots of 
any prime /^, = 22"+l>5. Also 7^/^^+1 = (mod F^+, >5). 
E. I. Grigoriev^"^ noted that a primitive root of a prime p can not equal 
a product of an even number of primitive roots [evident]. 
G. Wertheim^°^ treated the problem to find the numbers belonging to 
the exponent equal to the 1. c. m. of m, n, given the numbers belonging to 
the exponents m and n, and proved the first theorem of Stern. ^^ He dis- 
cussed (pp. 251-3) the relation between indices to two bases and proved 
(pp. 258, 402-3) that the sum of the indices of a number for the various 
primitive roots of w = p" or 2p" equals ^4){m)4> ]0(w) \ ■ If « belongs to the 
exponent 45 modulo p, the same is true of p — a (p. 266). He gave a table 
showing the least primitive root of each prime < 6200 and for certain larger 
primes; also tables of indices for primes < 100. 
P. Bachmann^°^ gave a generalization (corrected on p. 402) of Stern's^^ 
first theorem. 
G. Arnoux^°^ constructed tables of residues of powers and tables of 
indices for low composite moduli. 
A. Bindoni^°^ noted that a table showing the exponent to which a belongs 
modulo p, a prime, can be extended to a table modulo N by means of the 
following theorems. Let a, 61,..., &„ be relatively prime by twos. A 
number belonging to the exponent ti modulo bi belongs modulo 6162 ■ ■ -K 
to the 1. c. m. of ^1, . . . , ^^ as exponent. If ti is the least exponent for which 
a'''+l=0 (mod bi) and if the ti are all odd, the least t for which a'+l is 
divisible by 6], ... , 6„ is the 1. c. m. of ^i, . . . , i„. If p is an odd prime not 
dividing a and if a belongs to the exponent t modulo p, and a' = pg+l, and 
if p" is the highest power of p dividing q, then a belongs to the exponent 
lpn~i-u jjiojuio p". Hence if a is a primitive root of p, it is one of p" if 
s'Memorie Pont. Ac. Nuovi Lincei, 18, 1901, 261-348. 
^""Math. Quest. Educat. Times, 1, 1902, 90. 
"i/6id., pp. 108, 116. 
"^Kazani Izv. fiz. mat. obsc, BuU. Phys. Math. Soc. Kasan, (2), 12, 1902, No. 1, 7-10. 
"'Anfangsgrunde der Zahlenlehre, 1902, 236-7, 259-262. 
ii^Niedere Zahlentheorie, 1, 1902, 333-6. 
"'Assoc. fran§. av. sc, 32, 1903, II, 65-114. 
"«I1 Boll, di Matematica Giorn. Sc. Didat., Bologna, 4, 1905, 88-92. 
