702 HISTORY OF THE THEORY OF NUMBERS. [CHAP. XXIII 



G. B. M. Zerr 205 found six positive integers Xi such that each diminished 

 by f (zH ----- \-x 6 ) 5 becomes a fifth power. 



Several 206 found three numbers in arithmetical progression whose sum 

 is a sixth power. 



E. Swift 207 proved that x = 0, y 1250a 6 give the only integral solution of 



x+y 



= D. 



A. Cunningham 208 discussed X 2n +y 2n +z 2n = u' in +v* n +w 4n (n = l, 2) by 

 use of the identity a 4 +6 4 +(a+6) 4 = 2(a 2 +a&+& 2 ) 2 . Employ the usual 

 solution of u 2 + v 2 = w 2 , and set a; = u 2 v 2 uv, y = 2uv, z = x+y. Then 



He 209 expressed two special sextics and two octics in the form Y 2 qxZ 2 , 

 where Y, Z are functions of x, and <? = 17, 13, 19, 2. 



A. Gerardin 210 discussed the solution of x m +Ay p =f 2 , x m Ay p = g 2 . 

 Thus 2x m =f 2 +g 2 , so that x is & sum of two squares. 



Gerardin 211 treated the system x 6 l=4yz, Sy 3n l=xt, by taking as 

 x, t the factors 2y n 1, 4y 2n +2y n +l in either order, or t = l, or, fory = 2, 

 x = 2 k -l or 2 2fc +2 fc +l where n = k-l. 



E. N. Barisien 212 noted that x lz = r*'+s*-& = u*-v*-w* for r = 9i/ 4 , 

 s = x 4 +9x?/ 3 , i = 3x 3 7/+9?/ 4 , where u, v, w are sextic functions of x, y. 



A. Cunningham 213 noted that if N m =x m y m , and m, n are primes both 

 of the form 4&1, we can set 2V m =C = Fnwi, N n = tl=Fmul, simultaneously. 

 He and R. F. Davis 214 proved that we can express (x u +x 7 -}-l)l(x 2 +x+l) 

 in the forms A 2 +3B 2 and C 2 +7D 2 . 



Cunningham 215 investigated N = <j>(x, y} = <f>(x r , y'} = -, where 



4(x, y}=x fi y n x a y m 



and x } y are relatively prime integers. 



A. Gerardin 216 gave solutions of the system 



L. Aubry 217 made P(x+y}+Qx and P(x+y)+Qy both nth powers. 



206 Amer. Math. Monthly, 5, 1898, 114. 

 208 Ibid., 8, 1901, 48-9. 



207 Ibid., 15, 1908, 110-1. Problem proposed by J. D. Williams in 1832. 



208 Math. Quest. Educ. Times, (2), 14, 1908, 66-7 (reprinted, Mess. Math., 38, 1908-9, 102-3). 



209 Ibid., (2), 16, 1909, 105-6. 



210 Assoc. franc, av. sc., 37, 1908, 15-17. 



211 Sphinx-Oedipe, 6, 1911, 141-2. 



212 L'interm<diaire des math., 19, 1912, 194. Cf. Gerardin 88 of Ch. XXI. 



213 Math. Quest. Educ. Times, (2), 23, 1913, 21-22. 



214 Ibid., (2), 23, 1913, 86-8. 



216 Mess. Math., 44, 1914-5, 37-47. 



118 L'interm6diaire des math., 21, 1914, 143^; 24, 1917, 111-2. 



217 Ibid., 23, 1916, 33-4. Cf. Sphinx-Oedipe, 10, 1915, 26-27. 



