CHAP, vii] PEIMITIVE ROOTS, EXPONENTS, INDICES. 191 H. J. S. Smith47 stated that some of Oltramare's43 general results are erroneous at le.ast in expression, and gave a simple proof that xd^ 1 (mod pn) has exactly d roots if d divides 4>(pn). V. A. Lebesgue48 proved that, if p is an odd prime and a, b belong to exponents a, /?, there exist numbers belonging to the 1. c. m. m of a, £, as exponent. Hence if neither a nor $ is a multiple of the other, m exceeds a and ft. If d<p — 1 is the greatest of the exponents to which 1,..., p —1 belong, the latter do not all belong to exponents dividing d} since otherwise they would give more than d roots of #d==l (mod p). Hence there exist primitive roots of p. If a is odd, =±=1+2^ belongs to the exponent 2m~a modulo 2OT (p. 87). If h belongs to the exponent k modulo p, a prime, then h+Pz belongs modulo pn to an exponent which divides kpn~l (p. 101). If / is a primitive root of p, and fp~1 — l = pz} then/ is a primitive root of pn if and only if z is not divisible by p (p. 102). G. L. Dirichlet49 proved the last theorem and explained his21 system of indices for a composite modulus. V. A. Lebesgue50 published tables, constructed by J. Holiel,51 of indices and corresponding numbers for each prime and power of prime modulus <200, which differ from Jacobi's23 only in the choice of the least primitive root. There is an auxiliary table of the indices of x\ for prime moduli <200. V. A. Lebesgue62 stated that, if g<p is a primitive root of the prime p and if g' = gp~2 (mod p), then gf is a primitive root of p] at least one of g and gr is a primitive root of pn for n arbitrary. V. Bouniakowsky63 proved in a new way the theorems of Tchebychef34 that 2 is a primitive root of p = 8n+3 if p and 4n+l are primes, and of p = 4n+l if p and n are primes. He gave a method to find the exponent to which 2 or 10 belongs modulo p. A. Cayley54 gave a specimen table showing the indices a, j3,... for every number M = aa^. . .(mod A?"), where M< N and prime to N, for AT = 1,..., 50. There is no apparent way of forming another single table for all N's analogous to Jacobi's tables (one for each N) of numbers corresponding to given indices. F. W. A. Heime56 gave the least primitive root of each prune < 1000. His other results are not new. A secondary root of a prime p is one belonging to an exponent <p-~ 1 modulo p. 47British Assoc. Report, 1859, 228; 1860, 120, §73; Coll.1 Math. Papers, 1, 50, 158 (Report on theory of numbers). 48Introd. thforie des nornbres, 1862, 94-96. "Zahlentheorie, §§128-131,1863; ed. 2,1871; ed. 3, 1879; ed. 4, 1894. 60M6m. BOC. so. phys. et nat. de Bordeaux, 3, cah. 2, 1864-5, 231-274. "Formulas ct tables num6r., Paris, 1866. For moduli ^ 347,