202 HlSTOET OF THE THEORY OF NlJMBEBS. [CHAP. VII R. D. Carmichael120 called a number a primitive X-root modulo n if it belongs to the exponent X(n), denned in Ch. Ill, Lucas.110 The existence of primitive A-roots g is proved. The product of those powers of g which are primitive X-roots is = 1 (mod ri) if X(n) >2. A method is given to solve \(x) =a, and the solutions tabulated for ag24. C. Posse121 noted that in Wertheim's83' 91 table, the primitive root 14 of 2161 should be replaced by 23, while 10 is not a primitive root of 3851. E. Maillet122 described the manuscript table by Chabanel, deposited in the library of the University of Paris, giving the indices for primes under 10000 and data to determine the number having a given index. F. Schuh123 showed how to form the congruence for the primitive roots of a prime and gave two further proofs of the existence of primitive roots. He treated binomial congruences, quadratic residues and made applications to periodic fractions to any base. For any modulus n, he found the least m for which xm^l (mod n] holds for every x prime to n, and derived the solutions n of #(n) = ra, i. e., n's having primitive roots. F. Schuh124 discussed the solution of xqz=~ I (mod pa) with the least computation. If x belongs to the exponent q modulo n, the powers of x give a cycle of </>(q) numbers each with the "period" q. The numbers prime to n and having the period q may form several such cycles — more than one if n has no primitive root and q is the maximum period. If n = 2a (a>2), then # = 28 (s^a — 2) and the number of cycles is 1, 3 or 2 according as s = 0, s = l or s>l. In the last case, the cycles are formed by 2a~a(2&+l)=Fl. When q is even, x is said to be of the first or second kind according as 3.3/2== _ i (mod ri) or not. If the numbers of a cycle are of the second kind, we get a new cycle of the second kind by changing the signs of the numbers of the first cycle. While for moduli n having primitive roots there exist no numbers of the second kind, when n has no primitive roots and g is a possible even period, there exist at least two cycles of the second kind and of period q. Finally, there is given a table showing the number of cycles of each kind for moduli 5* 150. M. Kraitchik125 gave a table showing for each prime p< 10000 a primitive root of p and the least solutions of 2*=1, 10"= 1 (mod p). *J. Schumacher126 discussed indices. L. von Schrutka127 noted that, if q, r, . . . are the distinct primes dividing p — l} where p is a prime, all non-primitive roots of p satisfy 120Bull. Arner. Math. Soc., 16, 1909-10, 232-7. Also, Theory of Numbers, pp. 71-4. i21Acta Math., 33, 1910, 403-6. 122L'interm6diaire des math., 17, 1910, 19-20. Supplement de Vriend dor Wiskundc, Culwuborg, 22, 1910, 34-114, 166-199, 252-9; 25, 1913, 33-59, 143-159, 228-259. ™IUd., 23, 1911, 39-70, 130-159, 230-247. 126Sphinx-Oedipe, May, 1911, Numdro Special, pp. 1-10; errata listed p. 122 by Cunningham and Woodall. Extension to 25000, 1912, 25-9, 39-42, 52-5; errata, 93-4, by Cunningham. 126Blatter Gyrnnasial-RrhulwcBon, Miinchen, 47, 1911, 217-9. »7Monatshefte Math. Phys., 22, 1911, 177-186.