CHAP. XII] PELL EQUATION, a 2 +6z+c=n. 363 



Lagrange's reply has not been preserved, but it convinced Euler 78 of 

 the correctness of Lagrange's treatment of 101 = p 2 I3q 2 , though, being 

 then blind, Euler confessed he did not follow the real meaning of all the 

 deductions, nor the significance of all the letters introduced. 



Euler 79 noted that ar 2 4 = s 2 implies that ax 2 -f- 1 = y 2 holds for 



Thus, if a = 61, we may take r = 5, s = 39 and deduce the large numbers 

 x, y in his table. 



E. Waring 80 quoted results due to Brouncker and Euler. 



Euler 81 treated, essentially as had Brouncker, 49 an 2 + 1 = ?/ 2 , where a is 

 positive and not a square. Thus, for a = 5, y is > 2n and Euler set 

 y = 2n -f- p, whence n 2 = 4np + p 2 1, n = 2p + V5p 2 1. The radical 

 exceeds 2p, whence n > 4p. Set n = 4p + q, whence p 2 = 4pq + g 2 + 1, 

 p = 2q + V5g 2 + 1. Having now the initial radical, we may set q = 

 and obtain p = 1, n = 4, y = 9. For a = e 2 2 or e 2 1, we can give 

 explicit solutions n, ?/ : 



He repeated 82 his table 72 of the least positive solutions of an 2 + 1 = m 2 , 

 a < 100. 



Euler 83 treated / = a -f- bx + ex 2 = D as had Diophantus when a or c 

 is a square ; also the case in which / is a product of two linear functions, 

 /, ra of x, by equating / to the square of Ik, as well as the case in which / 

 equals I 2 + inn. In Ch. V, Euler noted certain forms which are never equal 

 to rational squares, as 3x 2 + 2, 3t 2 + (3ra + 2)w 2 , 5 2 + (5w d= 2)w 2 . In 

 Ch. VI, he noted that, given a/ 2 + bf + c = gr 2 , we can find new solutions of 

 ax 2 + bx + c = y 2 . Subtract and factor each new member; thus we may 

 set 



P(x-fi=q(y-g), q(ax+af+b)=p(y+g}. 



Multiply the first by p and the second by q and subtract. Hence 



aq 2 +p z 2pq 



. , 



x = ng-mf -- , y = mg-naf 2 bn, 



To obtain integral solutions take p 2 = aq 2 + 1 and change the sign of g. 

 Thus 

 x = 2gpq+f(aq 2 +p 2 } + bq 2 , y = g(aq*+p 2 )+2afpq+bpq, p 2 -aq 2 = l. 



The method for ax 2 + c = y 2 is similar, but simpler, giving x = qg + pf, 

 y = pg + a?/, and is derived a second way ( 86) given earlier by Euler. 66 



78 Opera postuma, I, 574; letter, March, 1770, to Lagrange, Oeuvres, XIV, 219. 



79 Ibid., 585; letter, Sept. 24, 1773, to Lagrange, Oeuvres, XIV, 239-40. 



80 Meditationes Algebraicae, 1770, 180-199; ed., 3, 1782, 308-337. 



81 Algebra, St. Petersburg, 2, 1770, Ch. 7, 96-111; French transl., Lyon, 2, 1774, pp. 116^ 



134; Opera Omnia, (1), I, 379-87. 



82 Also in Nova Acta Acad. Petrop., 10, ad annum 1792, 1797 (1777), 27; Comm. Arith., II, 



185. 



83 Algebra, II, Chs. 4-6, 38-95; French transl., 2, 1774, pp. 50-115; Opera Omnia, (1), I, 



349-78. 



