CHAP, xxvi] FERMAT'S LAST THEOREM. 775 



For reports on q u = (u p ~ l l)/p, see Ch. IV of Vol. I of his History. 

 There are additional notes by * E. Haentzschel 286 on 2 p ~ 1 = l (mod p 2 ), 

 p 1093, and H. E. Hensen 287 on the computation of q u . 



L. E. Dickson 288 gave an account of the history of Fermat's last theorem 

 and the origin and nature of the theory of algebraic numbers. 



F. Pollaczek 289 proved that, if x p +y p +z p = has integral solutions prime 

 to p, then q u is divisible by p if u si 31 for all primes p except a finite number; 

 also, x 2 -\-xy+y z = Q (mod p) is impossible. 



W. Richter 290 proved Korselt's 282 result for the special case m = n = r. 

 There exist rational integral functions x, y, z of t satisfying f=x n +y n +z n = 

 if and only if the genus \ (n l)(n 2) d r of the curve is zero, where d 

 is the number of double points and r the number of cusps. But d = r = Q 

 since dffdx 0, etc., hold only for x = y = z = 0. Hence n = 1 or 2. 



H. S. Vandiver 291 gave an expression for the residue modulo X n of Kum- 

 mer's 61 first factor hi of the number of classes of ideals in the domain 

 defined by a Xth root of unity. In terms of Bernoulli numbers we can 

 infer necessary and sufficient conditions that hi be divisible by any given 

 power of X. He 292 stated that if x p +y p +z p = Q holds for integers not 

 divisible by the prime p, then 23 P ~ 1 =1 (mod p 2 ) for p^l (mod 11), and 

 that the Bernoulli number B s is divisible by p 2 for s=(fp+l)/2, t=p 4, 

 p-6, p-S, p-10. 



A. Arwin 293 gave a method to solve (x+ l) p x p = 1 (mod p 2 ), p a prime. 



Vandiver 294 derived from one source the theorems of Furtwangler 249 and 

 the criterion of Kummer 76 for solutions prime to p of x p -\-y p = z p . 



P. Bachmann 295 gave an almost complete reproduction of the papers 

 by Abel, 16 Legendre, 17 Dirichlet, 20 Kummer, 61 Wendt, 152 Mirimanoff, 180 ' 246 

 Dickson, 195 - 6 - 199 Wieferich, 214 Frobenius, 228 - 238 and Furtwangler. 249 



Vandiver 296 employed the first factors hi and k of the class numbers of 

 the fields of the p n th and p n-1 th roots of unity respectively, and the value 

 of ki = hi/k due to J. Westlund, 297 and proved that ki is divisible by p if 

 and only if at least one of the first (p 3)/2 Bernoulli numbers is divisible 

 by p. Bernstein's 234 first assumption in his second case therefore implies 

 that p = l is a regular prime (so that his result forms no extension over 

 Kummer 61 ), while the assumptions in his first case do not as claimed in- 

 clude those of Kummer. 76 It is shown that 101, 103, 131, 149, 157 are 

 the only irregular primes between 100 and 167. 



286 Jahresber. d. Deutschen Math.-Vereinigung, 25, 1916, 284. 



287 L'enseignement math., 19, 1917, 295-301. 



288 Annals of Math., (2), 18, 1917, 161-87. 



283 Sitzungsber. Akad. Wiss. Wien (Math.) 126, Ho, 1917, 45-59. 



290 Archiv Math. Phys., (3), 26, 1917, 206-7. 



291 Bull. Amer. Math. Soc., 25, 1919, 458-61. 



292 Ibid., 24, 1918, 472. 



293 Acta Math., 42, 1919, 173-190. 



294 Annals of Math., 21, 1919, 73-80. 



195 Das Fermat Problem, Verein Wiss. Verleger, W. de Gruyter & Co., Berlin and Leipzig, 



1919, 160 pp. 



196 Proc. National Acad. Sc., May, 1920. 



297 Trans. Amer. Math. Soc., 4, 1903, 201-212. 



