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



quantity whose nth power is x-\-ty, and in giving the quantity the notation 

 p-\-eq had not specified the nature of p and q. 



Welsch 221 repeated a proof due to Catalan. 121 



D. Mirimanoff 222 considered the relation of F = x l -\-y l -\-z l = to cubic 

 congruences. Let x, y, z be the roots of t 3 Si 2 +s 2 s 3 = 0. Thus 

 F = <f>(si, 82, 83), where is a polynomial of degree I with integral coefficients. 

 We have si=0 (mod I). Let x, y, z be prime to I. By Legendre, 17 s[ F 

 is divisible by l(x+y)(x+z)(y+z) = Z($iS 2 s 3 ) ; call the quotient P(s\, s 2 , s 3 ). 

 Since SiS 2 s 3 is prime to I, and since sj is divisible by I 1 , F = Q gives 

 P(0, s 2 , s 3 ) = (mod /). Hence if F = has solutions prime to I, 



Z 3 +s 2 s 3 =0 (mod I), 



subject to P=0, has three roots. For Z = 3, then P = l and F = is im- 

 possible in integers prime to 1=3. For 1 = 5, P=s 2 ', but if s 2 =0, the 

 discriminant of the cubic congruence is 27 si, a quadratic non-residue of I, 

 so that it does not have three roots. The same argument applies to 1 = 11. 

 For 1= 17, the discriminant is a residue and there are three roots or no root; 

 the first case is excluded by the fourth criterion of Cailler (ibid., 10, 1908, 

 486; see p. 255 of Vol. I of this History) for cubic congruences. The method 

 fails for l = 3m+l, since we may now have s 2 ^0. 



Mirimanoff 223 employed Euler's expression for 1 2 P ~ 2 +3 P ~ 2 -?/ p ~ 2 

 as a polynomial in y to obtain a short proof of the final congruence used 

 by Wieferich to prove his criterion that 2 P-1 = 1 (mod p 2 ) . 



B . Lind 224 proved that x 2 + y z = z 6 is impossible in integers . Ifx n +y n =z n 

 is impossible, so are Z 2n Z 2 = 4F n and s(2s+l) = t 2n . The last equation 

 implies s = %", 2s+l = t 2 2 n , tit 2 = t, whence tf l = 2( 2 ) n , a case of Liouville's 32 

 equation. For a simpler proof, see Kempner. 281 



J. Westlund 225 noted that, if n is an odd prime, 



is divisible by n 2 if by n. Hence x n -\-y n = nz n is impossible if z is prime to n. 



R. D. Carmichael 226 proved that, if p and q are primes, p m q n =l 

 only for m=l, q = 2, p = 2 n +l; m = q = 2, n = p = 3; n = l, p = 2, q = 2 m 1. 



A. Fleck 227 distinguished cases A and B according as none or one (say x) 

 of the integral solutions =j=0 of x p -\-y p -\-z p = is divisible by the odd prime p. 

 Set s=x-\-y+z. Then 



(A) y+z = a p , z+x = b p , x+y = c p , s=abcp*GM, 



(B) y+z = p 2p - 1 a p , z+x = b p , 

 He considered the six quantities 



-zx = GK l 



221 L'interm6diaire des math., 16, 1909, 14-15. 



222 L'enseignement math., 11, 1909, 49-51. 



223 Ibid., 11, 1909, 455-9. Summary by Dickson, 288 p. 183. 



224 Archiv Math. Phys., (3), 15, 1909, 368-9. 



225 Amer. Math. Monthly, 16, 1909, 3-i. 



226 Ibid., 38-9. Special cases by G. B. M. Zerr, 15, 1908, 237. See Gerono. 92 



227 Sitzungsber. Berlin Math. Gesell., 8, 1909, 133-148, with Archiv Math. Phys., 15, 1909. 



