772 HISTORY OR THE THEORY OR NUMBERS. [CHAP, xxvi 



We may (pp. 402-3) replace this condition by the simpler one 264 



( s +l)p= s p-|-l (mod p 3 ), 



as noted by G. D. Birkhoff. The test fails for p = 6n-\-l since the con- 

 gruence has a root. He 2550 stated that x 6 y 6 =t= D 



N. Alliston 256 noted that x r y r =z m has integral solutions if r, m are 

 relatively prime positive integers. R. Norrie (pp. 33-4) treated the same 

 problem. 



R. Niewiadomski 257 considered d n = z n x n y n . If d n = for n an odd 

 prime, then d 2n +i is divisible by (x+y)(zx)(zy). He gave linear rela- 

 tions between d n+ i, d n , d n -\ and expressions for d n when di=0 (mod n k ) 

 and when d! 2 = 0. G. Me*trod (pp. 215-6) treated the latter case. 



E. Landau 258 noted that the assumptions 



X P-i=yir-i = l (mod p 2 ), x+y = mp, 



where p is an odd number > 1 not dividing m, lead to a contradiction. In 

 fact, 



1 = x*- 1 = (mp y) p - 1 ~(p l) mpy p ~ 2 + 1 (mod p 2 ) 



requires that p divide (p l}my p ~ 2 and hence also m. 



E. Miot 259 gave a false expression for the g.c.d. of 2 X 1, 3 X 1. 



H. Kapferer 260 proved Fermat's theorem for the exponents 6 and 10 by 

 showing by descent that 2 = (2 2 d=2/ 2 ) 2 (yz) z is impossible. 



H. C. Pocklington 261 noted that x 2n -{-y 2n = z 2 is impossible for all values 

 of n for which x n -\-y n = z n is impossible. For, if the former has solutions, 

 it has solutions with x prime to y and with y even. Thus x n = u 2 v 2 , 

 y n = 2uv. Hence u-\-v = a n , u v = {3 n and u, v equal 2 n ~ 1 y n , 8 n in some 

 order. Thus a n /3 n = (27) n . 



J. E. Rowe 262 proved that if x n +y n = z n , where x, y, n are odd, then x+y 

 is divisible by 2 n {^evident since the quotient of x n -\-y n by x+y is composed 

 of n terms and hence is odd]. From this main theorem II' we obtain his 

 theorem I' by changing the sign of y. 



Ph. Maennchen 263 reported on the history of the theorem. Several 

 (p. 294) proved that 2 n +l is an exact power only for 2 3 +l = 3 2 . 



W. Meissner 264 proved that x p +y p = z p is impossible in integers not 

 divisible by the odd prime p if there exists no integer v<p for which 



(v+l) p v p =l (mod p 3 ), v 3 ^l (mod p) 

 [cf. Carmichael 255 ]; also if p = 3*2 m d=l or 3 fc 2 w ; also if p, but not p 2 , is a 



2660 Bull. Amer. Math. Soc., 20, 1913, 80. 



268 Math. Quest. Educ. Times,, new series, 23, 1913, 17-18. 



267 L'intermddiaire des math., 20, 1913, 76, 98-100. 



268 Ibid., 206. 



259 Ibid., 112. Error noted pp. 183-4, 228. 

 260 Archiv Math. Phys., (3), 21, 1913, 143-6. 



261 Proc. Cambridge Phil. Soc., 17, 1913, 119-120. 



262 Johns Hopkins University Circular, July, 1913, No. 7, 35-40; abstract in Bull. Amer. 



Math. Soc., 20, 1913, 68-69. 



268 Zeitschr. Math. Naturw. Unterricht, 45J 1914, 81-93. 

 2M Sitzungsber. Berlin Math. Gesell., 13, 1914, 101-104. See Vol. I, Ch. IV, 39 of this History. 



