742 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxvi 



Let 6i(a), , e M _i(a) be units such that products of powers of them 

 multiplied by a m give all the units. Then 



A= 



It is shown that hi is divisible by X if and only if X divides the numerator 

 of one of the first (X 3)/2 Bernoullian numbers Bi = l/6, B 2 = l/30, ; 

 while if h z is divisible by X also hi is, but not conversely. He proved that 

 if X is not a divisor of H, condition (B) of Kummer 58 is satisfied. Hence if 

 X is an odd prime not dividing the numerator of any one of the first (X 3)/2 

 Bernoullian numbers, x +?/ x = 2 x is impossible in integers. 



The French Academy of Sciences 62 offered as a prize a gold medal of 

 value 3000 francs for a proof of Fermat's last theorem. After several 

 postponements of the date fixed for the award, the prize was finally (C. R., 

 44, 1857, 158) awarded to Kummer for his investigations on complex 

 numbers, though he had not been a competitor. 



Kummer 63 proved by use of prime ideals that, if X is an odd prime not 

 dividing the numerator of any one of the first (X 3)/2 Bernoullian numbers, 

 = has no solution in integers, nor in complex integers 



where a is an imaginary X-th root of unity. Thus there is no solution for 

 X<100, except perhaps for X = 37, 59, 67. This proof has been given hi 

 modern form, by use of Dedekind's ideals, by Hilbert. 153 

 A. Genocchi 64 proved that, if n is an odd prime, 



/77_1 \n- 



(V) 



and noted that this, in connection with a statement by Cauchy, 57 shows that 

 x n -\-y n -{-z n = Q is impossible in integers not divisible by the odd prime n 

 when n is not a divisor of the numerator of the Bernoullian number J5 (n _3)/ 2 , 

 the last one of the Bernoullian numbers in Kummer's condition. 



Kummer 65 noted that his assumption that B n is not divisible by X for 

 n= (X 3)/2 (as well as for smaller n's) corresponds to Cauchy 's 57 condition 



If not both #(x_3)/2 and B(\-6)/2 are divisible by X, one of the solutions x, y, z 

 of x +?/ x = z x must be divisible by X. Proof by Kummer, 76 pp. 61-5. 



62 Comptes Rendus Paris, 29, 1849, 23; 30, 1850, 263-4; 35, 1852, 919-20. There were five 



competing memoirs for the prize proposed for 1850 and eleven for the postponed prize 

 for 1853; but none were deemed worthy of the prize. Cf. Nouv. Ann. Math., 8, 1849, 

 362-3 and, for bibliography, 363^; 9, 1850, 386-7. 



63 Jour, fur Math., 40, 1850, 130-8 (93); Jour, de Math., 16, 1851, 488-98. Reproduced by 



Gambioli, 171 pp. 169-176. 



64 Annali di Sc. Mat. e Fis., 3, 1852, 400-1. Summary in Jour, fur Math., 99, 1886, 316. 



This congruence is a special case of one proved by Cauchy, M6m. Acad. Sc. Paris, 17, 

 1840, 265; Oeuvres, (1), III, p. 17. 

 86 Letter to L. Kronecker, Jan. 2, 1852, Kummer, 36 p. 91. 



