CHAP, vii] PRIMITIVE ROOTS, EXPONENTS, INDICES. 199 Fd(x) of xd—1 (dividing the last but not xl—I for t<d). To every divisor d of p-1 belong exactly <t>(d) numbers which are the roots of Fd(x)^Q (modp). P. G. Foglini" gave an exposition of known results on primitive roots, indices, linear congruences, etc. In applying (p. 322) Poinsot's9 method of finding the primitive roots of a prune 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 # is a residue of a square, (x3)6=l and #3 is the residue of a square. R. W. D. Christie100 noted that, if 7 is a primitive root of a prime p=4fc — l,the remaining primitive roots are congruent to p — y2" (n = 1,2,...) A. Cunningham101 noted that 3, 5, 6, 7, 10 and 12 are primitive roots of any prime Fn = 22"+1 > 5. Also Ff*+1=0 (mod Fn+1 > 5). E. I. Grigoriev102 noted that a primitive root of a prime p can not equal a product of an even number of primitive roots [evident]. G. Wertheim103 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,15 He discussed (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 m = pn or 2pn equals J<£(m)<£-j<£(m) [ • If 0 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. Bachmann104 gave a generalization (corrected on p. 402) of Stern's15 first theorem. G. Arnoux105 constructed tables of residues of powers and tables of indices for low composite moduli. A. Bindoni106 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,..., bn be relatively prime by twos. A number belonging to the exponent tt modulo 5; belongs modulo b$2. . .bn to the 1. c. m. of tit..., tn as exponent. If ^ is the least exponent for which a^+l=0 (mod &,-) and if the tt are all odd, the least t for which a'+l is divisible by b]}. .., 6n is the 1. c. m. of ti,.. ., tn. If p is an odd prime not dividing a and if a belongs to the exponent t modulo p, and al = pq-\-l} and if pu is the highest power of p dividing q, then a belongs to the exponent tpn~l~u modulo pn. Hence if a is a primitive root of p} it is one of pn if "Memorie Pont. Ac. Nuovi Lincci, 18, 1901, 261-348. 100Math. Quest. Educat. Times, 1, 1902, 90. lQlIbid.,'pp. 108, 116. 102Kazani Izv. fiz. mat. obsc., Bull. Phys. Math. Soc. Kasan, (2), 12,.1902, No. 1, 7-10.