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CHAP, vii] PRIMITIVE ROOTS, EXPONENTS, INDICES. 203 To this congruence he applied Hurwitz's42 method (Ch. VIII) of finding the number of roots and concluded that there are p 1 4>(p 1) roots. Hence there exist 4>(p l) primitive roots of p. A. Cunningham and H. J. Woodall128 continued to p< 100000 the table of Cunningham110 of the maximum residue indices v of 2 modulo p. C. Posse129 reproduced Korkine's118 and his own119 tables and explained their use in the solution of binomial congruences. C. Krediet130 treated x*=l (mod n) of Lucas,110 Ch. Ill, and called x a primitive root if it belongs to the exponent <p. The powers of such a root are placed at equal intervals on a circle for various n's. G. A. Miller131 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 =0 (mod m), and the sum of those belonging to the exponent 2 is = 1 (mod m), and proved the corresponding theorem by Stern15 for a prime modulus. A. Cunningham132 tabulated the number of primes p<104 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 106 for which y (y = 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 ykz= 1 (mod p) is solvable, where k is a given divisor of p 1. A. Cunningham133 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 £ modulo pk< 10000 for 2 = 2, rg 100; 2 = 3, 5, 7, 10, ll,rS30. J. Barinaga134 considered a number a belonging to the exponent g modulo p, a prime. If a is not divisible by 0, 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 if n<g} but to =F! if n = #, according as g is even or odd. A. Cunningham135 listed errata in his Binary Canon95 and Jacobi 's Canon.23 G. A. Miller136 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 27), but not modulo 27+1, where l<7</3, it belongs to the exponent 2&~y modulo 2'3. Also, if p is an odd prime, and N^== 1 (mod p), N belongs to the exponent p^~y modulo p0 if and only if N I is divisible by py, but not by py+l, where /3>7^ 1. «»Quar. Jour. Math., 42, 1911, 241-250; 44, 1913, 41-48, 237-242; 45, 1914, 1H-125. 129Acta Math., 35, 1912, 193-231, 233-252. l30Wiskundig Tijdskrift, Haarlem, 8, 1912, 177-188; 9, 1912, 14-38; 10, 1913, 40-46, 87-97. (Dutch.) 131Amcr. Math. Monthly, 19, 1912, 41-6. lMProc. London Math. Soc., (2), 13, 1914, 258-272. 133Mes8cnger Math., 45, 1915, 69. Cf. Cunningham.110 134Annacs Sc. Acad. Polyt. do Porto, 10, 1915, 74-6. "'Messenger Math., 46, 1916, 57-9, 67-8. "We/., 101-3.