CHAP, xxvi] FERMAT'S LAST THEOREM. 7G7 



and proved that (i) s has no factor other than a divisor of G in common 

 with one of these six expressions ; (ii) any two of the six have no common 

 factor other than a divisor of G, so that /, -, LI are relatively prime in 

 pairs; (iii) J, , LI are products of primes of the form Q^p+l; (iv) 



G. Frobenius 228 gave a simple proof of the criterion of Wieferich, 214 

 using Mirimanoff's 180 formulation of Kummer's criterion to show that 



A-l 



r, s=0 



is congruent modulo X, for every Z+0, 1, to both 



l-t c > 

 whence c^O (mod X), so that 2 X ~ 1 =1 (mod X 2 ). 



A. Gerardin 229 gave a brief history and extensive bibliography of the 

 subject. He conjectured that Fermat's last theorem could be proved by 

 showing that the difference or the sum of two nth powers (n > 2) is always 

 comprised between two consecutive nth powers. 



P. Bachmann 230 gave an account of results obtained by elementary 

 methods, chiefly those by Abel, 16 Legendre, 17 Wendt, 152 and Dickson. 195 " 9 

 The remark (p. 461) that all primes <100 are regular was corrected on 

 p. 480. 



H. Stockhaus 231 gave a lengthy exposition of known methods for expo- 

 nents 3, 5, 7, with suggestions of doubtful value on the general case. 



* K. Rychlik 232 gave a proof for exponents 3, 4, 5. 



* Ed. Barbette 233 proved some inequalities. 



F. Bernstein 234 proved Fermat's theorem under assumptions milder 

 than those of Kummer. 76 The second case (that in which one of the three 

 numbers is divisible by the prime exponent Z) is proved by means of the 

 assumption that the class number of the field k(Z) of the Z 2 th roots of unity 

 is divisible by Z, but not by Z 2 ; and again by means of the assumption that 

 k(Z) contains no class belonging to the exponent Z 2 , while the class number of 

 ^(r+T" 1 ) is prime to Z, where f z = l. The first case (that in which the 

 three numbers are prime to Z) is proved from the assumptions (i) that the 

 second factor /i 2 of the class number of &(f) is divisible by Z, and (ii) if Z" is 

 the highest power of Z dividing /i 2 , then in the " Teilklassenkorper" of the 

 Z"th degree every ideal of &(f), whose Zth power is a principal ideal in k(), 

 is itself a principal ideal. [See Vandiver's 296 criticisms.] 



228 Sitzungsber. Akad. Wiss. Berlin, 1909, 1222-4. Reprinted in Jour, fur Math., 137, 1910, 



314-6. 



229 Historique du dernier theoreme de Fermat, Toulouse, 1910, 12 pp. Extract in Assoc. 



frang. av. sc., 39, I, 1910, 55-6. All of his references are found in the present chapter. 



230 Niedere Zahlentheorie, 2, 1910, 458-476. 



231 Beitrag zum Beweis des Fermatschen Satzes, Leipzig, 1910, 90 pp. 



232 Casopis, Prag, 39, 1910, 65-86, 185-195, 305-317 (Bohemian). 



233 Le dernier theoreme de Fermat, Paris, 1910, 19 pp. 



234 Gottingen Nachrichten, 1910, 482^88, 507-516. 



