CHAP, vii] PRIMITIVE ROOTS, EXPONENTS, INDICES. 193 m modulo 6, and if bh is the highest psjwer of 6 which divides aw—1, and if ritZhy then bn divides ae—l where e=mbn~h. Further, if b is a prime*, a belongs to the exponent e modulo bn. For a new prime &', let m', n', h' have the corresponding properties. Then the exponent to which a belongs modulo B=bnb'n'... is the 1. c. m. L of m&n~A, m'Vn>-h>,.... For a = 10, we see that L is the length of the period for the irreducible fraction N/B. L. Sancery61 proved that if p is a prime and a<p belongs to the exponent B modulo p, there exists an infinitude of numbers a+px = A such that A9— 1 is divisible by pk, but not by pw, where k is any assigned positive integer. If A belongs to the exponent 6 modulo p>2, A will belong to the exponent B modulo pv if the highest power of p which divides Ae—l is i^p"; but if it be pv~*j A belongs to the exponent Qp* modulo p" [Barillari60*]. Hence A is a primitive root of p" if a primitive root of p and if Ap~l — 1 is not divisible by p2, and there are <£ ]0(p") \ primitive roots of p' or 2p". [Generalization of Arndt.33] C. A. Laisant62 noted that if a belongs to the exponent 3 modulo p, a prime, then a+1 belongs to the exponent 6, and conversely. If a belongs to the exponent 6, a+1 will not belong to the exponent 3 unless p = 7, a = 3. Hence if p is a prime 6ra+l, there are two numbers a, b belonging to the exponent 3, and two numbers a+1, b+1 belonging to the exponent 6; also, a+&=p—1. If (p. 399) p+# is an odd prime and p is even, then ppqq^qy p (mod p+g). G. Beiiavitis620 gave, for each power p'g 383 of a prime p, the periodic fraction for 1/p* to the base 2 and showed how to deduce the indices of numbers for the modulus p1. Let ? = pt~1(p —1) and let 2 belong to the exponent q/r modulo p\ A root b of br=2 (mod p1) is the base of the system of indices. G. Frattini63 proved by the theory of roo.ts of unity that, if p is a prime, the number of interchanges necessary to pass from 1, 2,..., p—2 to ind 2, ind 3,..., ind (p —1) and to ind 1-ind 2, ind 2-ind 3,..., ind (p-2)-ind (p-1) are both even or both odd. Fritz Hofmann64 used rotations of regular polygons to prove theorems on the sum of the primitive roots of a prime (Gauss7). A. R. Forsyth65 found the sum of the cth powers of the primitive roots of a prime p. The sum is divisible by p if p —1 contains the square of a prime not dividing c or if it contains a prime dividing c but with an exponent exceeding by at least 2 its exponent in c. If neither of these conditions is satisfied, the result is not so simple. "Bull. Soc. Math, de France, 4, 1875-6, 23-29. MM<Sm. Soc. Sc. Phys. et Nat. de Bordeaux, (2), 1, 1876, 400-2. 82aAtti Accad. Lincei, Mem. Sc. Fi8. Mat., (3), 1, 1876-7, 778-800. "Giornale di' Mat., 18, 1880, 369-76.