CHAP, xxvi] FERMAT'S LAST THEOREM. 745 



(i) The factor hi of the class number H is divisible by X, but not by X 2 . 



(ii) For ju, g, e(a) denned as by Kummer 61 , and for the integer v < (X 1)/2 

 such that B v =0 (mod X), there exists an ideal with respect to which as 

 modulus the unit 



k=0 



is not congruent to a Xth power, whence the second factor h z of H is not 

 divisible by X. 



(iii) The Bernoullian number B^ is not divisible by X 3 . 



All three conditions are satisfied when X = 37, 59, 67, the values <100 

 for which he had not previously proved Fermat's theorem. [But Kummer 

 (pp. 46-50) repeatedly used an earlier 760 congruence involving logarithms 

 which is not true in all cases, as noted by F. Mertens. 766 The remark that 

 this error vitiates also the present paper, and two further criticisms 

 were made by H. S. Vandiver. 76c First, Kummer (p. 42, bottom) relied 

 on his paper in Jour, de Math., 16, 1851, 473, where he reduced hi modulo 

 X, but not modulo X", n>l, as now needed. Second, Kummer (p. 53) 

 employed a decomposition of ty r (d) which holds only when it contains only 

 ideals of the first degree. Although the theorem on p. 61 is really subject 

 to this restriction, it is applied (p. 67) to ideals r (a) which are not proved 

 to be of the first degree. Kummer, 76a p. 120, had given the different de- 

 composition when there occur ideals not all of the first degree. J 



H. F. Talbot 77 proved (I) If n is odd >1, a n = b n +c n is impossible in 

 integers if a is a prime [Abel 16 ]]; (II) If n is any integer >l,andif a n = b n c n 

 is possible when a is a prune, then b c = l. For (I), (6+c) n >6 n +c n = a n , 

 b-\-c>a; b<a, c<a, b+c<2a. Hence b + c is not divisible by the prime a, 

 contrary to the given equation. Similarly for (II). Generalizations are 

 given. If a is a prime and m<n, a m = b n -\-c n is impossible if n is odd, while 

 a m = b n c n is impossible if b c>l. 



K. Thomas 78 attempted to prove Fermat's last theorem. 



H. J. S. Smith 79 gave numerous references on Fermat's last theorem, 

 noted that Barlow's 15 proof was erroneous, and reproduced the proof by 

 Kummer 63 for regular prunes. 



A. Vachette 80 proved (6) and concluded that, if a, b are integers and n 

 is a prime >2, (a-\-b) n a n b n is divisible by nab(a-\-b), and gave several 

 expressions for the quotient. Set 



A k = (x+l/x) k -x k -l!x k , a=x+lfx. 



Then A 6n +7 is proved divisible by (a 2 I) 2 [Cauchy 29 ]. There are proofs 

 (pp. 264-5) of (6) by induction on n and by Waring's formula. 

 F. Paulet 81 gave an erroneous proof of Fermat's last theorem. 



76a Kummer, Jour, fur Math., 44, 1852, 134 (error, p. 133). 



7f * Sitzungsber. Akad. Wiss. Wien (Math.), 126, 1917, Ha, 1337-43. 



76c Proc. National Acad. Sc., April, 1920. 



77 Trans. Roy. Soc. Edinburgh, 21, 1857, 403-6. 



78 Das Pythagoraische Dreieck und die Ungerade Zahl, Berlin, 1859, Ch. 10. 



79 Report British Assoc. for 1860, 148-152; Coll. Math. Papers, I, 131-7. 



80 Nouv. Ann. Math., 20, 1861, 160-6. 



81 Cosmos, 22, 1863, 385-9. Correction, p. 407, by R. Radau. 



