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



A. M. Legendre 21 stated that the discussion of (2), at least for special 

 exponents n, can be facilitated by a consideration of the cubic equation 

 whose roots are x,y,z; for integral roots, the discriminant must be a perfect 

 square. He was not entitled to conclude that x+y-\-z and xyz are divisible 

 by n 2 , as he had not proved that one of the unknowns is divisible by n. 



V. Bouniakowsky 22 argued that if x m +y m +z m = Q, where m is a prime 

 and x, y, z are integers with no common factor, and if N is chosen so that 

 m = 4>(N) l (which is possible for each prime m<31, except w = 13), 

 then xyz(xy+xz+yz) is divisible by N. But he used Euler's theorem 

 x *w i ( mo( j j\r) w hich is valid only when x is prime to N. 



Dirichlet 23 proved by descent that (2) is impossible in integers for n = 14, 

 also the impossibility of 



G. Libri 24 considered the number N z of sets of positive solutions <n 

 of z 3 +2/ 3 +l = (mod n), for a prime n = 3p+l. The equation for the 

 three periods of nth roots of unity is found in the form 



Comparing this with the known cubic, we get N 2 = n^La 2, where 



[Pepin 109 ]. Since a is comprised between zero and r= (4n 27) 1/2 , we have 

 Nz^nr2. Hence Nz increases indefinitely with n, and from a certain 

 limit on, z 3 +2/ 3 -fl = (mod n) is always solvable with neither x nor y 

 divisible by n. Having Nz, we can find the number of positive solutions <n 

 of x z +y 3 +u 3 +l=Q (mod n). 



If n is a prime 8w+l, so that n = a? + 16& 2 in a single way, the same 

 method of proof shows that the number of solutions of o; 4 +?/ 4 + 1 =0 (mod n) 

 is nQa 3, which increases with n. It is stated that one can prove 

 [Pellet, 128 - 244 Dickson 199 , Cornacchia, 217 Mantel 277 ] that a limit to the prime 

 p can be assigned such that, after passing it, the number of solutions of 

 x n +y n -\-l = Q (mod p) will always increase. Hence it is futile to try to 

 prove u n -\-v n = z n impossible by trying to show that one of the unknowns is 

 divisible by an infinitude of primes. 



E. E. Kummer 25 considered x 2x +y 2 *=z 2 *, where X is a prime, and x, y, z 

 are relatively prime by pairs. We may take y even. The third of four 

 possible cases is 



This is the only possibility if X = 8n+ 1, or if 2X+ 1 is a prime. If the initial 

 equation is solvable in integers, so is r 2x +s 2x = 2# 2x . As auxiliary to the 



21 The'orie dea nombres, ed. 3, II, 1830, art. 451, pp. 120-2; German transl., Maser, II, pp. 



118-120. 



22 Me"m. Acad. Sc. St. Pdtersbourg (Math.), (6), 1, 1831, 150-2. 



23 Jour, fur Math., 9, 1832, 390-3; Werke, 1, 189-194. Reproduced by Gambioli, 171 pp. 164-7. 

 u Jour, fiir Math., 9, 1832, 270-5. 



26 Jour, fur Math., 17, 1837, 203-9. 



