194 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vn J. Perott66 gave a simple proof that x^^l (mod pn) has pk roots. Thus there exists an integer b belonging to the exponent pn~~l modulo pn. Assuming the existence of a primitive root, of p, we employ a power of it and obtain a number a belonging to the exponent p 1 modulo pn. Hence ab is a primitive root of pn. Schwartz67 stated, and Hacken proved, the final theorem of Cauchy.14 L. Gegenbauer68 stated 19 theorems of which a specimen is the following: If p = 8a(8/3-}-l)4-24/3+5 and (p~l)/4 are primes and if 64a2+48a 4-10 is relatively prime to p, then 8a+3 is a primitive root of p. G. Wertheim69 gave the least primitive root of each prime < 1000 and companion tables of indices and numbers for primes < 100. He reproduced (pp. 125-130) arts. 80-81 of Gauss7 and stated the generalization by Stern.15 H. Keferstein70 would obtain all primitive roots of a prime p by excluding all residues of powers with exponents dividing p~l [Poinsot9]. M. F. Daniels71 gave a proof like Legendre's6 that there are <£(n) numbers belonging to the exponent n modulo p, a prime, if n divides p 1. *K. Szily72 discussed the "comparative number'1 of primitive roots. E. Lucas73 gave the name reduced indicator of n to Cauchy's26 maximum indicator of n, and noted that it is a divisor <</>(n) of <j>(ri) except when n=2, 4, pk or.2p*, where p is an odd prune, and then equals <t>(n). The exponent to which a belongs modulo m is called the "gaussien" of a modulo m (preface, xv, and p. 440). H. Scheffler74 gave, without reference, the theorem due to Richelot17 and the final one by Prouhet.32 To test if a proposed number a is a primitive root of a prime p, note whether p is of one of the linear forms of primes for which a is a quadratic non-residue, and, if so, raise a to the powers whose exponents divide (p 1)/2. L. Contejean75 noted that the argument in Serret's Algebre, 2, No. 318, leads to the following result [for the case a = 10]: If p is an odd prime and a belongs to the exponent e = (p I)/q modulo p, it belongs to the exponent pv~^e modulo p" when (ael}/p is not divisible by p, but to a smaller exponent if it is divisible by p [Sancery61]. P. Bachmann76 proved the existence of a primitive root of a prime p by use of the group of the residues 1,. .., p l under multiplication. "Bull, des Sc. Math., 9, I, 1885, 21-24. For fc = n-l the theorem is contained implicitly in a posthumous fragment by Gauss, Werke, 2, 266. 67Mathesis, 6,1886, 280; 7,1887, 124-5. 68Sitzungsber. Ak. Wiss. Wien (Math.), 95, II, 1887, 843-5. 69Elemcnte der Zahlentheorie, 1887, 116, 375-381. 70Mitt. Math. Gesell. Hamburg, 1, 1889, 256. 71Lineaire Congruences, Diss., Amsterdam, 1890, 92-99. 72Math. es termes 6rtesito (Memoirs Hungarian Ac. Sc.), 9, 1891, 264; 10, 1892, 19. Magyar Tudom. Ak. Ertesitoje (Report of Hungarian Ac. Sc.), 2, 1891, 478. 78The"orie des nombres, 1891, 429. 74Beitrage zur Zahlentheorie, 1891, 135-143. "Bull. Soc. Philomathique de Paris, (8), 4, 1891-2, 66-70. 76Die Elemente der Zahlentheorie, 1892, 89.