CHAP. XII] PELL EQUATION, ax 2 +fa:+c= D. 399 



1500<A^1700 the least solutions of x- Ay*=--l, when solvable, and 

 always those of x 2 -Ay-=+l. He noted (pp. 154-5) that the former is 

 solvable for 38 of the 110 composite numbers A = a 2 +6 2 between 1501 and 

 2000. Finally (pp. 162-190) he tabulated for 1500 < A ^2012 the period 

 and auxiliary numbers for the continued fraction for VZ [corresponding to 

 the first two lines in Degen's table]. 



R. Remak 297 modified Dedekind's 141 proof of the existence of solutions 

 of x z Dy' 2 = l and obtained upper limits on the least positive solutions: 



Known methods of solving y~ 2z- = 1 have been recalled. 298 



Th. Got 299 simplified the proofs by A. Meyer 150 . 



M. Simon 300 noted that Brahmegupta's first rule shows that he knew 

 how to solve all equations (a) 4(X 2 T2)z 2 +l = ?/ 2 and (b) (X 2 2)z 2 +l=;z/ 2 . 

 The identity (X 2 2)X 2 = (X 2 l) 2 -l gives x = \, ?/ = X 2 l for (b) and z = X/2, 

 y = \ 2 ^Pl for (a). But if X is odd, and a solution , j8 of (a) is found, it 

 becomes (/3 2 I)o; 2 /a 2 +l = ?/ 2 , which is satisfied if x = 2ap, whence the solu- 

 tion is z = X(X 2= Fl), ?/ = 2(X 2 Tl) 2 -l. 



G. Metrod 301 noted that in u*-2v 2 =l, v^2 a , a>l, and v^(2a) e . In 

 M 2 3t; 2 = l, v = 2 { only for t = 0, 2; v is not a power of an odd prime, and 

 0=1= (2o)', where a is an odd prime. In v? pv 2 = l, where p is an odd prime, 

 cases are noted in which v = 2* or a', where a is an odd prime =|=p. 



E. E. Whitford 302 extended Cayley's 137 table from D = 1000 to D = 1997, 

 but gave the solution of both x 2 Dy-= -4 and z 2 D?/ 2 =+4 when they 

 are solvable. He noted the application to finding the fundamental unit e 

 (least unit >1) of the domain defined by VZ>; the least positive solutions 

 of x*Dy* = l do not determine e when one of the equations x 2 Dy~= 1, 

 4 or 4 is solvable. 



0. Perron 303 obtains by use of continued fractions the limits 



for the least positive solutions of x z Dy 2 = l. Remak 297 had given larger 

 limits. Cf . Schmitz, 308 Schur. 314 



T. Ono 304 stated that, if z 2 -5i/ 2 = 4, 



1,1,1 T 2 o . T 2_o 



=--] --- 1 ----- r'"'> Xi x A, Xz Xii, '. 



2 x xx\ 



Infinite series involving successive solutions of this and x 2 Dy z = p* 

 have been treated. 305 



"V. G. Tariste" 306 noted relations between successive x's or y's for 

 which 



297 Jour, fur Math., 143, 1913, 250-4. Cf. Kronecker, 14 ' Perron. 303 



298 L'intermediaire des math., 20, 1913, 254-6. 



299 Annales Fac. Sc. Toulouse, (3), 5, 1913, 94-8. 

 3 Archiv Math. Phys., (3), 20, 1913, 280-1. 



301 Sphinx-Oedipe, 8, 1913, 137-8. 



302 Annals of Math., 15, 1913-4, 157-160. 



303 Jour, fur Math., 144, 1914, 71-73. 



3M L'intermediaire des math., 20, 1913, 224. 



305 Ibid., 21, 1914, 37-38, 47-48; 22, 1915, 21-23, 277-8. 



Ibid., 22, 1915, 125-6. 



