ON THE THEORY OF NUMBERS. 127 



1 +x\+x e 2 + . . . +4=0, mod \*. 



If Sj, S 2 , S 3 . . . denote the sums of the powers of the roots of the equation 

 F(y)=0, this formula may be written thus, — 



XNt=X*+S, +feS a +— =^ e 2 S,+ . . . e k S k+l , 

 or, solving for S v S 2 , . . ., 



e*S* + ,=x[Ni-ftN*_,+^=^N*_ 2 - ... -(-I/nJ-OX-I/. 



From this equation, when the values of N x , N 2 , &c, have been determined, 

 Sj, S 2 , . . . may be calculated, and thence by known methods tlis values of 

 the coefficients of the equation V(y)=0. Lastly, M. Lebesgue has shown 

 that, if we denote by o~ k the number of ways in which numbers divisible by 

 X can be formed by adding together k terms of the series y°, y l , . . y*-~ 2 , sub- 

 ject to the condition that no two powers of y be added the indices of which 

 are congruous for the modulus e, the function (X— l)F(y) assumes the form 



^C/-^/- 1 + ^ e - 2 -...+(-i) e ^]-(y-/) e t- 



But the practical application of any of these methods is very laborious 

 when X is a large number, chiefly on account of the determinations which 

 they all require of the numbers of solutions of which certain congruences are 



i— (— lyx 



susceptible. For e=2 the equation is y 2 +y-\ ^—r — - — =0, or, putting 



r=2y + \, r 2 —( — iy\=0. The cubic and biquadratic equations corre- 

 sponding to the cases e=3 and e=4 are also known from Gauss's investiga- 

 tions. The results assume the simplest forms if we put r=ey + 1 . We then 

 have 



(1) e=3, 4X=M 2 + 27N 2 , M=l, mod 3; r 3 -3Xr-XM=0. 



(2) e=4; X=A 2 + B 2 ; A=l,mod4; e=(-]/. 



[r' 2 + (l-2e)X] 2 -4X(r-A) 2 =0t. 

 Though these determinations are not required in M. Kummer's theory, we 

 have nevertheless given them here, in order to facilitate arithmetical verifi- 

 cations of his results. The forms of the period-equations for the case r=8 

 and e=12 can (it may be added) be elicited from the results given bv Jacobi 

 in his note on the division of the circle (Crelle, vol. xxx. pp. 167, 168.)« 



44. The Period- Equations considered as Congruences. — An arithmetical 

 property of the equation F(#)=0, which renders it of fundamental import- 

 ance in the theory of complex numbers, is expressed in the following theorem. 



"If q be a prime number satisfying the congruence qf=\, mod X, the 

 congruence F(j/)=0, mod q, is completely resoluble, i. e. it is possible to 

 establish an indeterminate congruence of the form 



F 0)=(y-«o) (V— U i) • • • (y—Ue-i), mod q, 



* In this congruence x lt x s ,... x k are k terms (the same or different) of a complete system 

 of residues for the modulus X ; and in counting the number of solutions, two solutions are to 

 be considered as different in which the same places are not occupied by the same numbers. 

 A simpler formula for S i+1 may be obtained by considering x Jt x 2 , ... x k to represent terms 



of a system of residues prime to X, and denoting by e k y k the number of solutions of M. Libri's 



congruence on this hypothesis. We thus find Sfc+i =Xy*— /* (Liouville, vol. iii. p. 116). 



t Liouville, vol. iii. p. 119. 



X M. Lebesgue, Comptes Rendus, vol. li. p. 9. Gauss has not exhibited this last equation 

 in its explicit form. See Theor. Res. Biq. /. c. 



