198 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vn ip the system of indices of n modulo k if fs of Kronecker.60 Such systems of indices y ..., v = pk~lqh~l..., where p, q,.. . are distinct i/ is not divisible by 4 or if JV=4, but € = 2 if AT is \et ^(AO denote the 1. c. m. of v/e, p — l,q — l,... maximum indicator for modulus AT]. For A .N). If N=pk, 2pk or 4 (so that 'N has primitive .j,s73]. There is a table of values of N<1000 and > iur which \fr(N) has a given value < 100. [ noted that we may often abbreviate Gauss' method root of a prime p by testing whether or not the trial root before computing the residues of all powers of a. pie rules to decide whether or not a is a quadratic or r* a is both a quadratic non-residue and a cubic non- d if af^l for every/ dividing p — I except f=p — l, t. 3«ve tables showing the residues of the successive _ divided by each prime or power of prime <1000, also i showing the indices x of 2X whose residues modulo pk are : tables are more convenient than Jacobi's Canon23 (errata ,he problem to find the residue of a given number with given power of a prime, but less convenient for finding all roots ._ jrder of a given prime. There are given (p. 172) for each power 000 of a prime p the factors of <t*(pk), the exponent £ to which 2 belongs .ulo pk} and the quotient <£/£. E. Cahen96 proved that if p is a prime >(32m+I-l)/2m+3 and if q = 2m+2p+l (m>0) is a prime, then 3 is a primitive root of q, whereas Tchebychef34 had the less advantageous condition p>32m+1/2wl+2. Other related theorems by Tchebychef are proved. There are companion tables of indices for primes < 200. G. A. Miller97 applied the theory of groups to prove the existence of primitive roots of pn, to show that the primitive roots of p2 are primitive roots of pn, and to determine primitive roots of the prime p. L. Kronecker98 discussed the existence of primitive roots, defined systems of indices and applied them to the decomposition of fractions into partial fractions. He developed (pp. 375-388) the theory of exponents to which numbers belong modulo p, a prime, by use of the primitive factor «Sitzungsber. Ak. Wien (Math.), 106, II a, 1897, 259. MNouv. Ann. Math., (3), 17,1898, 303. MMath. Quest. Educat. Times, 73, 1900, 45, 47. WA Binary Canon, showing residues of powers of 2 for divisors under 1000, and indices to residues, London/ 1900, 172 pp. Manuscript was described by author, Report British Assoc., 1895, 613. Errata, Cunningham.135 "Elements de la the*orie des nombres, 1900, 335-9, 375-390. 97Bull. Amer. Math. Soc., 7, 1901, 350. "Vorlesungen liber Zahlentheorie, 1,1901, 416-428.