138 REPORT 1860. 



* F 2n . | 



X— 1 



(i)- ' 



if we observe that (X being prime to y) the numbers I -, I — , ... I i^ '— 



7 7 7 



are congruous (mod X) to the fractions—-, — -, .... — - , taken in a cer- 



y 7 y 



tain order. But, by a curious property of the function F 8n _ 1 , demonstrated 



for the first time by M. Kummer, 



x=l \ 7/ 2ny*>-i 



The condition for the divisibility of H by X is therefore that one of the p 

 congruences included in the formula B„ (y 2n — 1) = 0, mod X, should be satis- 

 fied. The last of these congruences, or B^ (y 2 ^— 1)=0, is never satisfied; 

 for it is easily proved that the denominator of B^ contains X as a factor, 

 while y 2 ^— l = (y^ + l) (yn— 1), though divisible by X, is not divisible by X 2 . 

 And since, if w</x, y 2ra — 1 is prime to X, that factor may be omitted in the 

 remaining fi — ] congruences ; so that the condition at which we have arrived 

 coincides with that enunciated at the commencement of this article. 



We have exhibited M. Kummer's analysis of this problem with more ful- 

 ness of detail than might seem warranted by the nature of this Report, not 

 only on account of its elegance, but also because it exemplifies transforma- 

 tions and processes which are of frequent occurrence in arithmetical inves- 

 tigation*. 



52. "Exceptional" Primes. — A prime number X, which, like 37, 59, and 



\ a 



67 in the first hundred, divides the numerator of one of the first frac- 



2 

 tions of Bernoulli, and which consequently divides the number of classes of 

 ideal numbers composed with Xth roots of unity, is termed by M. Kummer 

 an exceptional prime. Such primes have to be excluded from the enunciation 

 of several important propositions ; and their theory presents difficulties which 

 have not yet been overcome. Thus the following propositions are true for all 

 primes other than the exceptional primes, but are not true for the exceptional 

 primes. 



(1.) The exponent to which any class of ideal numbers appertains (see 

 art. 49) is prime to X. 



(2.) The index of the lowest power of any unit which can be expressed 

 as a product of integral powers of the trigonometric units is prime to X. For 



that index is a divisor of — (see art. 42). 



(3.) Every complex unit which is congruous to a real integer for the 

 modulus X is a perfect Xth power. (Whether X be an exceptional prime or 

 not, the Xth power of any complex number is congruous, for the modulus X, 

 to a real integer, viz. to the sum of the coefficients of the complex number.) 



* In Liouville, vol. i. (New Series) p. 396, M. Kronecker has given a very simple demon- 

 stration of the congruence 



2n\^(y2n-i) = ( r 2»_l) [l2» + 22»+.. . + (X-l) 2 »],mod\2, 

 which, combined with another easily demonstrated formula, viz., 



1 2»+22» +. . (\-l)2» = ( - l)»-i B n X, mod X 2 [n < fi], 

 leads immediately to the theorem of M. Kummer. 



