266 report — 1S59. 



Hfl W) ' mod g ' 



£+1 7+1 



But also [F(p)] ?+1 =i» * Pi 3 ; 



£=? 2+1 /«\ 

 so that jo 3 _p x 3 = 1 — J , mod q ; 



or raising each side of this congruence to the power q — 1, 



Secondly, let ^ be a real prime of the form S«+ 1 ; we find 



[F(p)]^(^)V(p), mod q, or FCp)'- 1 ^^)', mod q ; 



g-i v-i 

 and also [F(p)] 3 ~ , =/> 3 Pl 3 . 



Hence ( — ) ( — ) =( ) > where q, is either of the complex factors of q; 



\7i/A?i/a \Pu3 



or, observing that f -?M = h^j , and(— ) =(— ) > we raa y write 



It is clear from this, that if we denote the four symbols ( — 1 , ( — ) , 

 /ca\ (El) by a,, ft,, b.,, a,, respectively, and the reciprocal symbols by 



a/, ft/, ft/, «/, we have the equations 



0^=0/6/, o 1 ft a =« ] 'ft/, a l a i =a 1 'aj=l, 

 «.&=«/*/, «,&.=«! 'ft ', ft.ft, = ft/ft/ = 1 , 



which imply that ur=dr\ ft = =ft-', &c., or, since «, a', ft, ft', . . • are cubic roots of 

 unity, 



(KW-a). 



V/iA \^i/a 



If juj and ^i be conjugate primes, the preceding proof fails; but it is easily 

 seen that in this case also 



\P%h XPJt 



Lastly, if p and q are both of the form 3» + 2, it follows from the defi- 

 nition of the symbols, and from Fermat's Theorem, that 



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