192 HlSTOBY OP THE THEOBY OF NUMBEBS. [CHAP. VII C. J. D. Hill56 noted that his tables of indices for the moduli 2n and 5n (n^5) give the residues of numbers modulo 10n, i. e., the last n digits. Using also tables for the moduli 9091 and 9901, as well as a table of logarithms, we are able to determine the last 22 digits. B. M. Goldberg57 gave the least primitive root of each prime < 10160. V. Bouniakowsky58 proved that 3 is a primitive root of p if p = 24n+5 and (p1)/4 are primes; 3 is a primitive root of p if p = 12n+ll and (p_l)/2 are primes; if p is a primitive root of the prime p=4n+l, one (or both) of p, pp is a primitive root of pm and of 2pm; 5 is a primitive root of p=20n+3 or 20u+7 if p and (p1)/2 are primes, and of p=40n+13 or 40n+37 if p and (p1)/4 are primes; 6 is a primitive root of a prime 24n+ll and 6 of 24n+23 if (p1)/2 is a prime; 10 is a primitive root of p=40n+7,19, 23, and 10 of p=40n+3, 27, 39, if (p1)/2 is a prime; 10 is a primitive root of a prime 80n+73, n>0, or 80n-f57, n>l, if (p1)/8 is a prime. If p = 8an+2a~l or 8an+a2 and (p1)/4 are primes, and if a?+l is not divisible by p, a is a primitive root of p. V. A. Lebesgue59 proved certain theorems due to Jacobi23 and the following theorem which gives a method different from Jacobi's for forming a table of indices for a prime modulus p: If a belongs to the exponent n, and if b is not in the period of a, and if / is the least positive exponent for which Z/=a*, then o/s=a has the root aV, where ft+iu l=m>; the root belongs to the exponent nf if and only if u is prune to /. Consider the congruence xm=a (mod p), where a belongs to the exponent n=(p--l)/n', and m is a divisor of n'. Every root r has a period of mn terms if no one of the residues of r, r2,..., rm~l is hi the period of a. If all the prune divisors of m divide n, the m roots have a period of mn terms; but if m has prune divisors g, r,..., not dividing n, there are only roots having a period of mn terms. The existence of primitive roots follows ; this is already the case if m=nr. Mention is made of companion tables in manuscript giving indices of numbers, and numbers corresponding to indices, constructed by J. Ch. Dupain in full for p<200, but from 200 to 1500 with reduction to one-half in view of ind pa=ind a=*=(p 1)/2 modulo p 1. L. Kronecker60 proved the existence of two series of positive integers 0& mi C7 = V "t P) such that the least positive residues modulo k>2 of Q-pQz*'$?* give all the <j>(k) positive integers <fc and prime to k, if ij = 0, 1,..., mi 1; t2=0, 1,..., m21; etc. [cf. Mertens92]. G. Barillari600 proved that, if a is prime to 6 and belongs to the exponent MJour. fiir Math., 70,1869, 282-8; Acta Univ. Lundensis, Lund, 1,1864 (Math.), No. 6, 18 pp. "Rest- und Quotient-Rechmmg, Hamburg, 1869, 97-138. "Bull. Ac. Sc. St. P&ersbourg, 14, 1869, 375-81. Comptes Rendus Paris, 70,1870, 1243-1251. "Monatsber. Ak. Berlin, 1870, 881. Cf. Traub, Archiv Math. Phys., 37, 1861, 278-94. ""Giornale di Mat., 9,1871, 125-135.