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CHAP, vii] PRIMITIVE ROOTS, EXPONENTS, INDICES. 187 where a, ft... are powers of distinct primes, use a as a factor in forming u in case a is prime to i\/a,, but as a factor of v in case a is prime to i3/a, and as a factor of either u or v indifferently in case a is prime to both ii/a and i'2/a. Since ii/u and i2/v are relatively prime indicators corresponding to bases m^ and m2v, it follows from the preceding theorem that the indicator corresponding to base m^-m^ and modulus n is ii 12 ifa , c • --------——— L c m< Of t ^ U V CO Hence, given several bases m^ 7n2,... and a single modulus n, we can find a new base relative to which the indicator is the 1. c. m. of the indicators corresponding to m^ ra2,.... If the latter bases include all the integers <n and prime to n, the corresponding indicators give all indicators which can correspond to modulus n, so that all of them divide a certain maximum indicator I. Then for every integer m relatively prime to n, m1 = 1 (mod n). If n = va} where v is an odd prime, or if n = 2 or 4, 7 = 0(n). If n = 24, fc>2, 7=</>(tt)/2. If Jy is the maximum indicator corresponding to a power % of a prime, and if n=TLnj} then I is the 1. c. m. of /!, 72> • • • • The equation mx—ny = l has the solution x^m1"1 (mod n). Cauchy27republished the preceding paper, but with an extension from the limit n = 100 to the limit n = 1000 for his table of the maximum indicator 7. C. F. Arndt28 gave (without reference) Gauss' second proof of the existence of a primitive root of an odd prime p, and proved the existence of the cj)(pn) primitive roots of pn or 2pn, and that there are no primitive roots for moduli other than these and 4. If t is a divisor of 2a~~2, n>2, exactly t numbers belong to the exponent t modulo 2H (p. 18). If, for a modulus pn, 2pn, a belongs to the exponent t, then a-a2. . .a' is congruent to ( — I)'"1"1 (pp. 26-27), while the product of the numbers belonging to the exponent t is congruent to +1 if t^2 (pp. 37-38). He proved also Stern's15 theorem on the sum of these numbers. He gave the same two theorems also in a later paper.29 L. Poinsot30 used the method of Legendre6 to prove the existence of <f>(n) integers belonging to the exponent n, a divisor of p —1, where p is a prime. He gave (pp. 71-75) essentially Gauss' first proof, and gave his own9 method of finding primitive roots of a prune. The existence of primitive roots of pn, 2p'1, 4, but of no further moduli, is established by use of the number of roots of binomial congruences (pp. 87-101). C. F. Arndt31 noted that if a belongs to an even exponent t modulo 2n, then =±=a, ^a3,..., ^a'"""1 give the t incongruent numbers belonging to the exponent t, and are congruent to ft • 2m =F 1 (/c = 1, 3,5,. . .). The product of the numbers belonging to the exponent t modulo 2n, n>2, is = +1. 27Exercicea d'Analyse et de Phya. Math., 2, 1841, 1-40; Oeuvrca, (2), 12. 28Archiv Math. Phys., 2, 1842, 9, 15-16. 29Jour. fiir Math., 31, 1846, 326-8. 30Jour. de Math6matiques, (1), 10, 1845, 65-70, 72. aiAv^itr Mo+v. PV.ITQ A I«AP; ^af; ^OQ