400 HISTORY OF THE THEORY OP NUMBERS. [CHAP, xn 



A. S. Werebrusow 307 stated erroneous conditions involving N = 

 for the solvability of x-Ny^= 1. 



Thekla Schmitz 303 proved that, for the least positive solutions of 

 x z Dy z = l, x+y-^D<2e iD ) where e is the base of natural logarithms. 



A. Cunningham 309 described and noted errata in various tables on the 

 Pell equation : Euler, 72 Legendre, 88 Degen, 101 Cayley, 137 and Bickmore. 219 



Kiveliovitchi 310 gave an elementary method of solving 6rc 2 +l=2/ 2 . 

 We may take x = 2u, y = 5u v, 2v = w. Then x = 5w+2r, y = 12w+5r, 

 r 2 = 6wJ 2 +l. Hence if (aji = 0, 2/1 = 1), , (Xi, y^, are the solutions 

 arranged in order of increasing magnitude, Xi + i = 5xi+2yi, yi+i = 12xi+5yi. 

 The same method is said to apply to ax 2 +l = y 2 if a = 4h-\-2, 4a+l = HI. 



A. Gerardin 311 applied the remark of Hart 174 on Ay* 1 = D, A = r 2 +s 2 . 

 To treat similarly x*-Ay z = 2, set A=a?-2b 2 , y = a*-2(3' 2 , and solve the 

 system of equations 



Thus, if A = 151 = 13 2 -2-3 2 , we get = 7, a = 59, ?/ = 3383, which leads to 

 Legendre's solution of x 2 151i/ 2 =l. For x 2 Ay 2 = 4, set A = a 2 -\-b' 2 , 

 y=z z -\-t z and solve the system 



(bz-aty-AP=2b, (bt+az} 2 -Az 2 =^2b. 



Thus, if A = 3 2 +10 2 , we get the least solution t = 3, 2 = 4. An error for 

 A = 397 in Legendre's 88 table is noted. He announced an extension in MS. 

 to 3000 of the table by Whitford. 302 



M. Cassin 312 gave relations between successive solutions of # 2 = 3?/ 2 -f-l. 



Several 313 gave relations between successive solutions of 2 2 Dz 2 =l 

 or c, and of ux 2 vy z = w. 



*J. Schur 314 obtained closer limits than had Remak, 297 Perron, 303 and 

 Schmitz. 308 



On x z -3y z = l, see papers 100 of Ch. I; 12, 24, 29, 33, 51 of Ch. V; 94 

 of Ch. VII; 230 of Ch. XXI. On 2z 2 l = D, see papers 112-129 of Ch. 

 IV; 92 of Ch. XXIII. For ax*+by* = c or ax-+bxy+cy* = k, see Ch. XIII. 

 On 5z 2 rb4= D, see Wasteels 72 of Vol. I, p. 405. On the application to fac- 

 toring, see Vol. I, p. 368. For "Pell equations of higher order," see papers 

 313-23 of Ch. XXI, 19-25 of Ch. XXIII, and Ch. XXVI. Pell equations 

 occur incidentally in the following papers: 56, 70, 107, 152, 178, 185, 187, 

 189, 196, 202, 204, 210, 219, 223, 227 of Ch. I; 135 of Ch. IV; 41, 109 of 

 Ch. V; 138, 193 of Ch. VI; 66 of Ch. XV; 55 of Ch. XVI; 21 of Ch. XVII; 

 270-4 of Ch. XXI; 111, 250 of Ch. XXII; 95, 99, 163 of Ch. XXIII. 



807 L'interm&iiaire des math., 22, 1915, 202-3; 23, 1916, 56 for admission of errors. 

 *8 Archiv Math. Phya., (3), 24, 1916, 87-9. Cf. Perron. 303 

 309 Mess. Math., 46, 1916, 49-69. 



810 Soc. Math, de France, Comptes Rendus Stances, 1916, 30-1. 



811 Sphinx-Oedipe, 12, June 15, 1917, 1-3; 1'enseignement math., 19, 1917, 316-8; 1'inter- 



m^diaire dea math., 24, 1917, 57-58. 



812 L'intermSdiaire des math., 25, 1918, 28, 93. 

 tl3 Ibid., 83-87; 26, 1919, 51-54. 



814 Gottingen Nachrichten, 1918, 30-6 



