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



v=zy, w = x+yz]. Let x n +y n = z n , and set x=zu, y = z v. Then 

 z n -n(u+v)z n ~ l + (tf+v^z*- 2 



Hence u n -\-v n is divisible by z. Similarly, a = u n +(v u} n is divisible by x, 

 and P = v n -\-(u v) n by y. His argument that Fermat's equation is im- 

 possible if n is odd and > 3 is unsatisfactory. By Cotes' theorem, 



u n -\-v n = (u-{-v)Ii(u 2 2uv cos XTT/n+v 2 ), 

 where X = l, 3, 5, , n 2. The Xth quadratic function has the factors 



w+^=t2 Vwy cos \7r/(2n). 

 But u n +v n has the factor z = u+v+w, whence 



w = 2 -Juv cos X7r/(2?i) . 

 Similarly, since a is divisible by x = u+ (v u) +w, and /3 by y = u+ (u v) +w, 



X 7T / ; ~ - v X 7T 



( ji n I \ pn<3 tn = / \7M ii ?)) crm 



V U (Aj I L-LJo _. , (AJ ^/ c/^M/ t/y x^VJio 



ju?Z 



whereas the one is real and the other imaginary. He also claimed that the 

 first w is symmetrical in u, v, while the third w is not. He made also the 

 error of assuming that an even factor of a product of an odd by an even 

 number must divide the latter. 



J. A. Grunert 73 proved that, if n>l, there are no positive integral values 

 satisfying x n -\-y n = z n unless x>n, y>n, simultaneously. Set z=x-\-u and 

 apply the binomial theorem; hence y n >nx n ~ 1 u. 



L. Calzolari 74 considered a triangle whose sides are integral solutions of 

 x n +y n =z n , nodd >1. Thus z i =x i axy+y*^P 2 ioT a suitable value of a. 

 It is stated that the polynomial P n ^x n +y n is divisible by P 2 , the polynomial 

 quotient P n _ 2 is divisible by P 2 , etc., and finally the symmetric quotient 

 Pi = #+2/ equals z, which is impossible. If n = 2m, P=P n , a = 0, m = l. 



G. C. Gerono 75 considered the integers x, y for which a x b v =l for 

 primes a, b. If a > 2, then b 2, a = 2 n + 1, and x = 1, y = n when n>l, with 

 also x = 2, y = 3 when n = l. If a = 2, then b = 2 n 1, rc = n, ?/ = !. . 



E. E. Kummer 76 proved that for any relatively prime integral solutions 

 of z x +?/ x =2 x , where X is any odd prime, and xyz is prime to X, 



(8) B ( i_ i}lz Pi(x, y} = (mod X) (t = 3, 5, - , X-2), 



where B } is the jth Bernoullian number and P,(rc, ^/) is the homogeneous 

 polynomial of degree i for which 



/ 

 V 



dv i /=(, (, 



He proved that Fermat's equation is impossible in integers for odd 

 prime exponents which satisfy the following three conditions : 



73 Archiv Math. Phys., 26, 1856, 119-120. Wrong reference by Lind, 241 p. 54. 



74 Annali di Sc. Mat. e Fis., 8, 1857, 339-345. 

 76 Nouv. Ann. Math., 16, 1857, 394-8. 



Abh. Akad. Wiss. Berlin (Math.), for 1857, 1858, 41-74. Extract in Monatsb. Akad. Wiss. 

 Berlin, 1857, 275-82. 



