CHAP, vii] PEIMITIVE BOOTS, EXPONENTS, INDICES. 197 J. Perott85 employed the sum sk of the kth powers of 1, 2, . . . , p — 1, and gave a new proof that ^==0, . . ., v~2=0, V-i= —1 (mod p). If w is the 1. c. m. of the exponents to which 1, 2, . . ., p — 1 belong, evidently sm==p — 1, whence m>p— 2. If A belongs to the exponent m, then A, A2, . . ., Am are incongruent, whence wgp-1. Thus A is a prirnitive root. N. Amici86 proved that, if v>2, a number belongs to the exponent 2"~2 modulo 2" if and only if it is of the form 8/1=*= 3, and called such numbers quasi primitive roots of 2". For a base 8/1=*= 3, numbers of the two forms Sk+1 or 8k=*=3, and no others, have indices. The product of two numbers having indices has an index which is congruent modulo 21"""2 to the sum of the indices of the factors. The product of two numbers &i and b2, neither with an index, has an index congruent modulo 2"~2 to the sum of the indices of —61 and — 62- The product of a number with an index by one without an index has no index. K. Zsigmondy87 proved by use of abelian groups that, if 8 = qikl , . .qrkr, m=pi7ri. . .p/*, where #1, . . ., qr are distinct prunes, and PI, . . ., pa are distinct prunes, the number of incongruent integers belonging to the exponent 5 modulo m is where 5y is the g. c. d. of 5 and tj=4>(pfj)j while Zt- is the number of the integers h, . . . , ta which contain the factor qf*. E. de Jonquieres88 proved that the product of an even number of primitive roots of a prime p is never a primitive root, while the product of an odd number of them is either a primitive root or belongs to an exponent not dividing (p — 1)/2. Similar results hold for products of numbers belonging to like exponents. Certain of the n integers r, for which rn is a given number belonging to the exponent e = (p — l)/n, belong to the exponent ne, while the others (if any are left) belong to an exponent ke} where k divides n. He conjectured that 2 is not a primitive root of a prime p=l, 7, 17 or 23 (mod 24); 3 not of pssl, 11, 13 or 23 (mod 24); 5 not of pssl, 11, 19, or 29 (mod 30). These results and analogous ones for 7 and 11 were shown by him and T. Pepin89 to follow from the quadratic reciprocity law and Gauss' theorems on the divisors of z2=*= A. G. Wertheim90 added to his84 corollaries cases when 6, 10, 11, 13 are primitive roots of primes 2g+l, 4#-f-l; also, 10 is a primitive root of all primes 8#+l 5^137 for which g is a prune 10&+7 or 10&+9, and of primes 16g+l for which q is a prime 10&+1 or 10&+7. Wertheim91 gave the least primitive root of each prime between 3000 and 5000 and of certain higher primes. He noted errata in his83 table to 3000. 85Bull. des Sc. MatWmatiques, 18, I, 1894, 64-66. 86Rendiconti Circolo Mat. di Palermo, 8, 1894, 187-201. "Monatshefte Math. Phys., 7, 1896, 271-2. 88Comptes Rendus Paris, 122, 1896, p. 1451, p. 1513; 124, 1897, p. 334, p. 428. "Comptes Rendus Paris, 123, 1896, pp. 374, 405, 683, 737.