CHAP, xxvi] FERMAT'S LAST THEOREM. 773 



divisor of a number of one of these four forms ; and if p~ divides one of the 

 four forms, provided k and m are divisible by p. 



The congruence 5 x +7"+ll r =0 (mod 13) was solved by several writers. 265 



T. Suzuki 266 found the 12 sets of solutions of 5*+8"+l l z =0 (mod 13). 



L. Aubry 267 noted that, if m is prime to n, x m +y m = z n has the solution 

 x = A u a, y = A u b, z = A v , where nvmu = l, a m +b m = A. For m = 3, n = 2, 

 he gave a solution involving two parameters. 



A. Ge>ardin 267a gave integral solutions of x 3 y*=z n for 2^n^S. 



H. S. Vandiver 268 wrote q(r) for (r p ~ l l}fp and proved that if 



x p +y p +z p = 



is satisfied by integers not divisible by the prime p, then 



J)E=0 (modp) 



is satisfied by each of the six values t = x/y, -, zfy, and either g(2)=0 

 (mod p 3 ), g(3)=0 (mod p), or else g(2)=g(3)=g(5)==0 (mod p) and, if 

 p = 2 (mod 3), g(7)=0 (mod p). 



E. Swift 269 proved that neither of x 6 y 6 is a square. 



H. S. Vandiver 270 proved that if x p +y p +z p = Q is satisfied in integers 

 prime to p, then g(5) = (mod p) and 1 + f + H -+l/Q)/5]s=0 

 (mod p). 



G. Frobenius 271 proved that, if Fermat's equation has integral solutions 

 each prime to the prime exponent p, then q(m) is divisible by p for ra = ll 

 and m=l7, and, in case p = 5 (mod 6), also for m = 7, 13, 19. Moreover, 



vanishes identically modulo p for m ^22 and m = 24, 26. Here the symbolic 

 power h* is to be replaced by the Bernoullian number & x . 



J. G. van der Corput 272 proved the impossibility of x 5 +?/ 5 = Az 5 for A = 1 

 and other values. 



R. Guimaraes 273 gave a bibliography and discussed the history of 

 Fermat's last theorem, including Wronski's 66 pretentions. 



N. Alliston 274 proved that Fermat's theorem for odd exponents implies 

 that &4rH-2_|_ c 4n+2 = Q j s impossible if w>0. 



265 Math. Quest. Educ. Times, new series, 26, 1914, 101-3. 



266 Tohoku Math. Jour., 5, 1914, 48-53. Further report in Ch. XXIII. 105 



267 L'interme'diaire des math., 21, 1914, 19-20. 



267a Sphinx-Oedipe, 9, 1914, 136-9. For 7 3 -10 2 =3 s , ibid., 6, 1911, 91. 



268 Trans. Amer. Math. Soc., 15, 1914, 202-4. 

 269 Amer. Math. Monthly, 21, 1914, 238-9; 23, 1916, 261. 



270 Jour, fur Math., 144, 1914, 314-8. 



271 Sitzungsber. Akad. Wiss. Berlin, 1914, 653-81. 



272 Nieuw Archief voor Wiskunde, 11, 1915, 68-75. 



273 Revista de la Sociedad Mat. Espanola, 5, 1915, No. 42, pp. 33-45. There is a great 



number of confusing misprints. Both Crelle's Journal and Comptes Rendus Paris are 

 cited as C.r., the second being once cited as Cr., Berlin! 



274 Math. Quest. Educ. Times, new series, 29, 1916, 21. 



