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Full text of "History Of The Theory Of Numbers - I"

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. ri   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.