264 report — 1859. 



are so chosen that the norms of p 2 , p 3 . . . form a continually decreasing series 

 (as is always possible); lastly, the units i* are so chosen as to render p 2 , p 3 ... 

 primary. Let p s =a e +ia s ; let — %(a s — 1)=0 S > mod 4 ; and in the series 

 0!, 2 • . . d n +i, let p be the number of sequences of uneven terms. Then 



VP2/4 



t -2p + 2V # 



Example. Let it be required to determine whether the congruence 

 ar 4 = — 3381, mod 11981 be possible or impossible. 



Since 1 1 981 = 109 2 + 10 2 , and is a prime number, the resolubility of this con- 

 gruence depends on that of the congruence # 4 = — 3381, mod (— 109+10£). 



qqqi \ 



i on 1 1 n • )• This 



gives us the series of equations 



— 3381 e=(31+3z)(— 109+100+* 8 (-17+28»), 



-109 + 10i=(2 +'2i)(- 17 +28i)+i°(-19-12i), 



— 17 +28z'= — 2« (— 19 — 12t)+«°(+ 7 — 10t), 

 -19— 12t= — 2* ( 7 — 10i)+?(— 1 — 2*')i 



7 — 10t'= (3 + 50 (— 1 — 2 +«*• 



Here 0!=— 1, 2 = + l, 3 =2, 4 =1, 0=1 ; so that p=% S/i0=O, and 

 ( — uToXTTv) = ^' or * ne P ro P ose d congruence is resoluble. Its four roots 



are +87, +2646, as may be found by any of the indirect methods for the 

 solution of Quadratic congruences. 



37. Cubic Residues. The Theory of Cubic is less complex than that of 

 Biquadratic Residues, and is at the same time so similar to it, that it will not 

 be necessary to treat it with the same detail. 



If/» be a real prime of the form 3« + l, and if l,f,f 2 denote the roots of 

 the congruence x 3 — 1=0, mod p, the p — 1 residues k lt k 2 . . . k p _i of p divide 

 themselves into three classes according as k^~ ==lj or =/, or =f 2 , mod p; 

 the first class comprising the cubic, the two other classes comprising the 

 non-residues. Now it can be proved that every prime number of the form 

 3« + l may be represented by the quadratic form A 2 — AB + B 2 ; i. e. it may 

 be regarded as the product of two conjugate complex numbers of the forms 

 A + Bp, A+Bp 2 , where p and p 2 are the two imaginary cube roots of unity; 

 just as the theory of biquadratic residues involves the consideration of the 

 quadratic form A 2 + B 2 , and of complex numbers of the type A + Bi. The 

 real integer A 2 — AB + B 2 is the norm of the complex numbers A + Bp 

 and A + Bp 2 , and expresses the number of terms in a complete system of 

 residues for either of those modules. 



The theory of these complex numbers has not been treated of in detail by 

 any writer (see Eisenstein, Crelle, vol. xxvii. p. 290); but the methods of Gauss 

 or Dirichlet are as applicable to them as to complex numbers involving i. 



Thus it will be found that every fraction of the form „ can be developed 



in a finite continued fraction, having for its quotients complex integers; that 

 Euclid's process for finding the greatest common divisor is applicable in this 

 case also, and that the same arithmetical consequences may be deduced from 



