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



A. M. Legendre 17 remarked that the French Academy of Sciences had 

 offered one of its prizes for a proof of Fermat's last theorem, but without 

 awarding the prize. He considered x n +y n +z n = Q for n a prime >2 and 

 for relatively prime integers x, y, z each 4= 0. He noted ( 3, 4) that 

 x+y+z is divisible by n, and its nth power by (x+y)(y+z)(z+x) } by a 

 proof criticized and completed by Catalan. 91 Let 



be the quotient of y n +z n by y+z. Then ( 7) y+z and < have the g.c.d. 

 n or are relatively prime according as x is or is not divisible by n. 

 First, let x be divisible by n. Then ( 8, 10) 



1 



y+z=-a n , <f>(y, z)=na n , x=-aa, 



m 



x+y = c n , <f)(x,y)=y n , z=-c-y, 



where a is an integer divisible by n, and each prime factor of a, /3 or 7 is 

 of the form 2kn+l. Each prime factor of a is of the form 2tn 2 +l ( 11), 

 and x, assumed divisible by n, is divisible by n 2 ( 13), both results being 

 credited to Sophie Germain in the foot-note to 22. 



Second, let no one of the numbers x, y, z be divisible by n. Methods 

 applicable only in the special cases n = 3, 5, 7, 11, but not to n = 13, etc., 

 are given in 14-20. To Sophie Germain is credited the proof ( 21-22) 

 that, if n is an odd prime <100, 



(2) x n +y n +z n = Q 



has no integral solutions each prime to n. This proof is called " very 

 ingenious, quite simple, and of an almost absolute generality." As noted 

 above, y+z is prime to <f>(y, z), and their product equals ( x) n ; hence we 

 may set 



y+z = a n , <f>(y, z)=a n , x=-aa, 



(3) z+x = b n , <j>(z,x)=(3 n , y=-bp, 



x+y = c n , <f>(x,y)=j n , z=-cy, 

 whence 



(4) 2x = b n +c n -a n , 2y = a n +c n -b n , 2z = a n +b n -c n . 



THEOREM. If there exists an odd prime p such that 



(5) -j-77 w +^0 (mod p) 



has no set of integral solutions , TJ, f , each not divisible by p, and such that 

 n is not the residue of the nth power of any integer modulo p, then (2) 

 has no integral solutions each prime to n. 



For, if x, y, z are integers satisfying (2), they satisfy congruence (5), 

 so that one of them, say x is divisible by p. Then, by (4), 



a) n =0 (modp). 



17 Sur quelques objeta d'analyse ind6termin<5e et particulierement sur le thdoreme de Fermat, 

 M6m. Acad. R. Sc. de Tlnstitut de France, 6, anne"e 1823, Paris, 1827, 1-60. Same, 

 except as to paging, The'orie des nombres, ed. 2, 1808, second supplement, Sept., 1825, 

 1-40 (reproduced in Sphinx-Oedipe, 4, 1909, 97-128; errata, 5, 1910, 112). 



