CHAP, xxvi] FERMAT'S LAST THEOREM. 741 



-4 



the odd prime n if 1+2n _ 4+3n _ 4+ , /n-l\ 



is not divisible by n [i. e., if the Bernoullian number Z?< B _ 3 )/ 2 is not divisible 

 by n], or if a certain number w (p. 359) is prime to n. Cf. Genocchi, 64 

 Kummer. 65 



E. E. Kummer 58 proved that z x ?/ x = 2 x is impossible for the series 59 

 of real primes X for which (A) the number of non-equivalent ideal complex 

 numbers formed from an imaginary Xth root a of unity is not divisible by 

 X and (B) every complex unit E(a), which is congruent modulo X to a 

 rational integer, equals the Xth power of another complex unit. These two 

 conditions are satisfied if X = 3, 5, 7, but probably not for X = 37. 



G. L. Dirichlet 60 noted that Kummer's condition (A) relates to a theory 

 closely analogous to the fact that a number m for which D is a quadratic 

 residue is not always represented by x 2 Dy z , but by one of several quadratic 

 forms, and similarly for the forms in X 1 variables defined by norms of 

 complex integers based on a. 



Kummer 61 proved that, for the domain defined by an imaginary Xth 

 root a of unity, where X is an odd prime, the number of classes of ideals is 

 the product H = hih 2 of the two integers 



P D 



where n = (X 1)/2, and P, D, A are defined as follows. Let /? be a primitive 

 root of iS*" 1 = 1, and g a primitive root of X. Then 



- 1 ), 0(0) = 

 where </ is the least positive residue of g i modulo X. Next, 



is a complex unit (a divisor of 1). Then, if Ix denotes the real part of log x } 



le(oi) le(a a ) 



68 Berichte Akad. Wiss. Berlin, 1847, 132-9. 



59 "I prove that it is impossible for an infinitude of primes X, but do not know for just which 



X's the assumptions hold." That these X's are infinite in number was believed, but not 

 proved, by Kummer. He called the remaining primes exceptional (as 37, etc.). The 

 same statements were made in 1847 in letters to Kronecker (Kummer, 36 pp. 75, 84). 

 In his Vorlesungen liber Zahlentheorie, 1, 1901, 23, Kronecker stated that Kummer 

 proved the impossibility of x x +r/ x =2 X for an infinitude of primes X and at first believed 

 that his proof applied to nearly all X's, but later believed the contrary. Kummer, 35 p. 

 32, is elsewhere quoted as believing it probable that there are approximately as many 

 regular primes as irregular (exceptional) primes. A. Wieferich, Taschenbuch fur 

 Mathematiker u. Physiker, Leipzig, 2, 1911, 108-111, stated that Kummer proved 

 Fermat's last theorem for an infinite series of exponents. 



60 Berichte Akad. Wiss. Berlin, 1847, 139-141; Werke, II, 254-5. 



61 Berichte Akad. Wiss. Berlin, 1847, 305-319. Same in Jour, fur Math., 40, 1850, 93-138; 



Jour, de Math., 16, 1851, 454-498. 



