CHAP. XII] PELL EQUATION, ax 2 +bx+c=n. 385 



W. P. Durfee 187 stated that, if X Q , y Q ; x 1) y l ; be the integral solutions 

 of ax- y-= 1, arranged according to magnitude, then 



x n y n+t x n +ty n = x t , ax n x n +ty n yn+t = yt- 



S. Giinther 188 noted that the solution of 2x 2 l=y 2 was apparently 

 known to Plato. Its complete solution implies that of 2x 2 +l=y 2 , and 

 conversely. To solve (a 2J [-b 2 )x 2 l = y 2 , seek the integral solutions of 

 2 -(a 2 +6 2 )77 2 = a 2 and test whether or not a 2 divides 2(a 2 +& 2 )7y 2 =b26^; 

 if so we have a solution of the initial equation. 



E. de Jonquieres 189 found the period of the continued fraction for VA 

 for special types of numbers A, and treated periodic continued fractions 

 whose numerators differ from unity. 



D. S. Hart 190 stated that a process, simpler than Euler's and Lagrange's, 

 to find integral solutions of ax 2 +bx+c=\I\ is to subtract such a square 

 (lx-\-m) 2 that the difference will factor into two linear functions with integral 

 coefficients. Then L 2 -\-MN- D = (LMrfs") 2 gives x; equate its denomi- 

 nator to unity. 



E. Catalan 191 discussed Ax 2 = y 2 -j-l. Thus A is of the form a 2 +6 2 . If 

 p, q give the least solution, x is divisible by p. Set x = pz; then 



Hence consider (a 2 +l)x 2 = 7/ 2 +l. For its solutions, 



It is shown that x n is a sum of three squares if n^3. If 6> 1 in the initial 

 equation, x n is a sum of four squares. Every integer y>l, for which 

 (a 2 +l) 2 = 2/ 2 1, is a sum of three squares. Cf. Catalan 63 of Ch. VII. 



S. Roberts 192 proved that q 2 Dr 2 = l can be solved by using the nearest 

 integral limits exclusively or superior limits exclusively as the partial 

 quotients belonging to the continued fraction for VZ>, instead of using the 

 customary inferior limits exclusively. But he admitted his results are due 

 to Stern 145 and Minnigerode. 159 



G. de Longchamps 193 gave a bibliography of Pell's equation. 



J. Perott 194 proved that there exists a positive integer X such that, in 



w x is divisible by a given odd prime, where ti, Ui give the least positive 

 solutions of t 2 du 2 = l. He repeated (pp. 342-3) Poincare's 185 remark. 



M. Weill 193 noted that x 2 -Ay 2 =N 2 has the solution x = Au 2 +t 2 , y = 2tu, 

 if t, u give a solution of t 2 Au? = N. Taking N=l, consider a, a it a 2 , -, 



7 Johns Hopkins University Circular, 1, 1882, 178. 



188 Blatter fur Bayer. Gymnasialschulwesen, 18, 1882, 19-24. 



189 Comptes Rendus Paris, 96, 1883, 568, 694, 832, 1020, 1129, 1210, 1297, 1351, 1420, 1490, 



1571, 1721. 



190 Math. Magazine, 1, 1882-4, 40-1. 



191 Assoc. franc.. av. sc., 12, 1883, 101; Atti Accad. Pont. Nuovi Lincei, 37, 1883-4, 84-95. 



192 Proc. London Math. Soc., 15, 1883-4, 247-268. 



193 Jour, de math, elem., 1884, 15 (1885, 171, on continued fractions). 



194 Jour, fur Math., 96, 1884, 335-7. 



195 Nouv. Ann. Math., (3), 4, 1885, 189-193. 

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