CHAP, xii] PELL EQUATION, ax*+bx+c= D. 383 



A. Martin 173 noted that, in the least solution of 2 9817?/ 2 = 1, x has 97 

 digits. 



D. S. Hart 174 noted that (r 2 +s 2 )?/-l = D for ?/ = ra 2 +n 2 , if 



where one of r, s is odd and the other even, while n is to be found by trial. 



Martin 1740 found the least solution of x 2 -978ly 2 = 1. 



S. Roberts 175 noted that if P Du-= 1 is solvable in integers, where 

 D = 2 lt a a j3 b - - , /i = or 1 and a, J3, odd, then P-D f u~= -1 is solvable, 

 where D' = 2 fl a a+2p (3 b+2q - -. Since any prime 4ri+l is a D, any odd power 

 of it is a D'. If D = s 2 d, the solvability of P du 2 = 1 is a necessary, but not 

 sufficient, condition for the solvability of t z Du-= 1. 



Roberts 176 proved that, if t, u are the least solutions of P Au? = l, there 

 are values h, HI, less than t, u, for which either Mt\ Nu\ = 1, MN = A, 

 or Mi\Nu\ = . c 2, MN = A, unless M = l. If the first of these equations 

 ig solvable and M<N, then M is the middle denominator of the period of 

 the continued fraction for VZ; but if the second holds, and not the first, 

 2M is the middle denominator. 



H. Brocard 177 gave a bibliography and historical notes on Pell's equation. 



K. E. Hoffmann 178 recalled that Lagrange proved that x , y Q is a solution 

 of ' Ay 2 = l if x o/?/o is the convergent corresponding to the first or first 

 two periods of the continued fraction for VA. Other solutions follow from 



While it is usually merely stated that x n /y n is a convergent to a later com- 

 plete period, a direct proof is here given by use of the u closed form " of 

 a periodic continued fraction (ibid., 62, 1878, 310-6). 

 A. Kunerth 179 gave a " practical " method of solving 



(17) y z = ax 2 +bx+c. 



If a rational solution is known, we may transform (17) into 



(18) y*=(ax+W+(yx+d)(6X+fi. 



Hence every such transformation yields two values 5/7 and f/e of x 

 giving rational solutions. If x = mfn, y = r/n is a solution of (17), take 

 7 = n, 6= m. Then r = ma-}-nj3 ) from which we may determine a, /3. 

 Then c, f may be found from (18). To proceed without a known solution, 

 subtract (az+/3) 2 from (17) and employ the condition that the difference 

 be a product of two linear functions : 



(19) (6 



173 The Analyst, Des Moines, 4, 1877, 154-5. 



174 /bid., 5, 1878, 118-9. 



1740 Math. Visitor, 1, 1878, 26-7. 



176 Proc. London Math. Soc., 9, 1877-8, 194. 



176 Ibid., 10, 1878-9,30-32. 



177 Nouv. Corresp. Math., 4, 1878, 161-9, 193-200, 228-232, 337-343. 



178 Archiv Math. Phys., 64, 1879, 1-8. 



179 Sitzungsber. Akad. Wiss. Wien (Math.), 78, II, 1878, 327-37. 



