CHAP, xxvi] FERMAT'S LAST THEOREM. 735 



Hence a, b, or c is divisible by p. But if 6 were divisible by p, then, by (3), 

 y=b(3 would be divisible by p, and hence by (2) also z would be divisible 

 by p, whereas x, y, z have no common factor. Similarly, c is not divisible 

 by p. Hence 



a^O, x=0, z=-y, <f>(x, y}=y n ~ 1 , <f>(y, z)=ny n ~ 1 (mod p). 



Thus, by (3), y n =y n ~ 1 , a n =ny n ~ l . Hence ny n = a n (mod p}. By the 

 final equation (3), 7 is prime to p. Hence we can determine an integer 71 

 such that 771 = 1 (mod p). Thus n= (ayi) n (mod p), contrary to hypothesis. 



The theorem applies if n = 7, p = 29, since the residues of the seventh 

 powers modulo 29 are 1, 12, no two of which differ by unity, and no 

 one of which is congruent to 7. Similarly, for each odd prime n < 100, S. 

 Germain gave a p for which the theorem applies. 



The condition that n shall not be a residue of an nth power requires 

 that p be of the form mn+1, where evidently m is even. Legendre proved 

 ( 23-28) that m must be prime to 3 and that both conditions in the 

 theorem hold if p = mn+l is a prime and m = 2, 4, 8, 10, 14, 16 (but over- 

 looked the exceptional character of n = 3 when m = 14, 16; cf. Dickson 195 ). 

 He concluded that (1) has no solutions prime to n for n an odd prime < 197. 



He proved 18 ( 38-47) that x 5 +y 5 +z 5 = has no integral solutions and 

 that if solutions of (2) exist for n = 7, 11, 13 or 17, they involve a great 

 number of digits ( 29-37). 



Schopis 19 argued that, if x 5 y 5 = w & , where xyw is prime to 5, then 



x-y = u 5 , 

 and 



x*+x 3 y-i \-y* = u 



is a fifth power, say (u 4 -\-z) 5 . Thus 

 5yA =z(5u l6 +lQu u z+Wu 8 z z +5u*z 



Thus z is divisible by 5 and the second member by 25. Thus A is divisible 

 by 5, which is seen to be impossible. 



G. L. Dirichlet 20 proved that there are no relatively prune integers 

 x, y such that x 5 y 5 = 2 m 5 n Az 5 , m and n being positive integers, n={=2, 

 and A not divisible by 2, 5 or a prime Wk+1. With the same restrictions 

 on A, the theorem holds also if n = 0, m^O, and 2 m A = 3, 4, 9, 12, 13, 16, 

 21, or 22 (mod 25). If n>0, n=(=2, and if A is not divisible by 2, 5 or a 

 prune 10&+1, there exist no relatively prime integers x, y such that 

 z 5 7/ 5 = 5 n A,z 5 . The last shows that x*y 5 =z 5 is impossible in integers 

 (since one of the unknowns, say z, must be divisible by 5); the proof is 

 analogous in the two cases z even and z odd, whereas Legendre 18 employed 

 two methods. 



18 This proof was reproduced in Legendre'a The'orie des nombres, ed. 3, II, 1830, arts. 654- 



663, pp. 361-8; German transl. by H. Maser, 1893, 2, pp. 352-9. If 2 is the unknown 

 divisible by 5, the proof for the case z even is like Dirichlet's, 20 while that for z odd is 

 by a special analysis. 



19 Einige Satze Unbest. Analytik, Progr. Gumbinnen, 1825, 12-15. 



" Jour, fur Math., 3, 1828, 354-375; Werke I, 21-46. Read at the Paris Acad. Sb., July 11 

 and Nov. 14, 1825 and printed privately, Werke, I, 1-20. Cf . Lebesgue." 



