136 REPORT— 1860. 



Lim - 





L(2\)' i - 1 



when X increases * without limit. It will be seen that the number of classes 



of ideal numbers for X=3, X=5, X=7, is unity; i.e., for those values of X 



every complex prime is actual. In the absence of any determination of a 



system of fundamental units for X=ll, X=13, X = 17, and X=19, it is not 



possible to say whether this is or is not the case for these values also. But 



p 

 from and after the limit X=23, the value of the factor ; — r indicates 



that a complex number is not necessarily a complex prime because it is 

 irresoluble into factors. 



51. Criterion of the Divisibility of H by X. — The number of classes of 

 ideal numbers, which we have symbolized by H, is not in general divisible 



by X ; but in certain cases it may happen that it is so. The quotient — is 



p 



never divisible by X, except when the other factor - — — — is also divisible 



by X. And it has been found by M. Kummer that the necessary and sufficient 



p 

 condition for the divisibility of 7 ~-^ by X is that the numerator of one of 



(2xy~ i J 



the first p.— 1 fractions of Bernoulli should be divisible by X. The investi- 

 gation of this singular criterion depends on a transformation of the function 

 4>(/3) which enters into the expression of P. If we represent the product 

 (y/3-l)^(/3) = (y rA _ 2 -l) + ( y _y 1 ) / 3 + (yy 1 - r2 )/r-+....+(yy^ 3 - 

 y\-2)P K ~ 2 > hi which every coefficient is divisible by X, by 



KK + b ] fl + h(i 2 +...b k _ f/- 2 2, or X^K/3) 



(b m denoting the quotient y V m -}~ ym , or I 72 m r\ if I represent the greatest 



X X 



integer contained in the fraction before which it is placed), we obtain by 

 multiplication the equality 



(/+l)P=X^(/3)^(/3 3 )....^(^- 2 ); 

 or, since y^ + l is divisible by X, and may be supposed not divisible by X 2 f, 



C denoting a coefficient prime to X. The congruence — - — =0, mod X, 



D (2X>*-» 



is therefore equivalent to the congruence 



,K/3)-K/3')....M3 x - 2 )=0,modX, 

 which may, in its turn, be replaced by the following, 



»Ky)»Ky 3 )...»Ky*- 2 )=0,modX. 

 For, if there be an equation which, considered as a congruence for a given 

 modulus X, is completely resoluble for that modulus, any symmetrical function 

 of the roots of the congruence is congruous, for the modulus X, to the cor- 

 responding function of the roots of the equation. The function ^(/3) i£(/3 3 ) 



* Liouville, vol. xvi. p. 473. The formula is given without demonstration. 



t For y^+l and ( y +\)f + l are hoth of them divisible by X ; but only one of them can 

 be divisible by X 2 , since their difference is not divisible by \ 2 . We can therefore, without 

 changing y y l ... y A _ 2 , determine y in accordance with the supposition in the text. 



