1^4 CF THE THEORY OF NUMBERS. [CHAP, vii r * * tf jf^ ! nmti p), If p is a prime >2, he concluded (p. 165) that p is i f *<*W i i th* the radical signs in the formula for a primitive r* *»*•'<!< -f unity. Cf. of Ch. VIII. I1*, j »fi»"* the subject. ^ _ )/2_ I I\ *»r> ' * a primitive root of a prime p satisfies x(p ^ = — 1, I *t », t i;»» <*f tin* ^==-1 (mod p), *=(p-l)/(2a), where a -,4/%2' * ivr tb wld of p — 1; while a number not a primitive r«*;« '*t *t)r«* at one of the x*s — 1. Hence if each a'^ — 1 and • - - ! » <i ia a primitive root. V, \ Ijttt^gu?*3 that prior to 1829 he had given in the Bulletin 4n N. r«i Mt*enw, the congruence X=0 of Cauchy14 for the integers In i' t*i the n modulo p. I I 'ji^hy*4 the existence of primitive roots of a prime p, essen- f fj Jj 445 in I proof. If p — 1 is divisible by n = cfWc* . . . , where * \p ure primes, he proved that the integers belonging to the \ p coincide with the roots of F <*»-l)(a^- !)(**•-!). . . 1 •nF"-i)(^-i)...(±—-i)...s*° (mod p)' 1 fjf f v f* ni the1 Z = 0 are the primitive nth roots of unity. For tfo* iil^^e a* visor n of p— 1, the sum of the Zth powers of the primitive r v*- »»i x* = I p) is divisible by p if I is divisible by no one of the n, it/12, n/6, . . . , n/afe, . . . , n/abc, .... P * if -rvcral of are divisors of Z, and if we replace n, a, by . . . by C ^ !, 1-a^l— fc, , , in the largest of these divisors in fractional form, we 4 I u fn-'Vn congruent to the sum of the Zth powers. In case xm^l * I i hi# IF* distinct integral roots, the sum of the Zth powers of all the "^ :« > modulo p to m or 0, according as I is or is not a multiple ^ f\ 1 \ .1 - * M L ^tern:i proved that the product of all the numbers belonging to * f nt i i^ m 1 (mod p), while their sum is divisible by p if d is divisible j'^r* , but is s ( - 1)11 if d is a product of n distinct primes (generaliza-f t iri'i^ D. A., arts. 80, 81). If p = 2n+l and a belongs to the expo-i. t> c product of two numbers, which do not occur in the period of a, - ;n t! e period of a. To find a primitive root of p when p - 1 = 2db . . . , -. !>, are distinct odd primes, raise any number as 2 to the powers /»- 1 ) 6, . . . ; if no one of the residues modulo p is 1, the negative LTt of these residues is a primitive root of p] in case one of the > l^use 3 or 5 .in place of 2. If p = 2g+ 1 and $ are odd primes, 2 as p = 8n+3 or 8n+7. If p = . -r / ,,c- P poljtechmque, cah. 18, t. 11, 1820, 345-410. >.n** r fnt to Eccydopaedia Britannica, 4 1824 698 < -• fie Mith , 2,1S37, 25S. ' 4j sw .^ 2 %Iath, 1829, 231; Oeunres, (2), 9, 26d, 278-90. J*, a- J»* Mith 6, 1830, 147-153.