CHAP. Xlll] SOLUTION OF ax 2 +by 2 = c. 407 



and noted that the resulting equation in d has real roots if 



27 



which is a consequence of 2X 2 +2Y 2 = 



SOLUTION OF 

 L. Euler 48 noted that 



He 49 noted that, if m 2 = a&n 2 +l, then ax 2 -by 2 = af 2 -bg 2 for 



x V^+T/ V& = ( / Va+gr V&) (m+n V^6) x . 



C. F. Kausler 50 treated the solution of m'x 2 +riy 2 = N, where 

 w' = 4m-fl, n' = 4n+2. Thus x = 2X+l, y = 2Y, whence 



Let >4m+l and set = (4m+l)D+#. Then 



(2 



*-' z ) & 



n 



2 

 Since (2n+l) j&= (4m+l)g has the solutions 



the question is whether i = D = F 2 . If so, we test (10 by the table of pronic 

 numbers X(X+1) in Nova Acta, XIV, 253. A similar treatment is given 

 for the case ra / = 4m 1, n' = 4n+l. 



C. F. Gauss 51 solved mx 2 +ny 2 = A by the method of exclusions. 



F. Arndt 52 noted that, if/, h are given relatively prime integers, the least 

 solutions offp 2 hq 2 =k, k = 1 or 2, can be found, without using continued 

 fractions, by means of the least solutions of x 2 fhy z = l, given in Table X 

 of Legendre's Theorie des nombres (errata noted, p. 246). We have only 

 to take x = Tl+2/p 2 /fc, y = 2pq/k. He gave a table of the least roots of 

 p0=- p '0'2 = l or 2 for 3^pp'^1003. 



S. Realis 53 solved (ft +4) a; 2 ny 2 = 4 by formulas simpler than those 

 given by the usual method of employing a Pell equation. If a, j3 give a 

 solution, then 



x = |[(n+2)a+n/r], y = *[(n+4)a+ (n+2)/3] 



give a second solution. We thus get an infinitude of sets of solutions 

 (1, 1), (1+n, 3+w), , which are said to give all. Replacing x by 2u+l, 

 y by 2v+l, we get (n+4)(tt 2 +^) =n(v--\-v). Hence the above work solves 

 the problem to find an infinitude of pairs of triangular numbers whose 

 ratio is n : n+4. 



48 Opera postuma, 1, 1862, 490 (about 1769). 



49 Ibid., 215 (about 1774). 



50 Nova Acta Acad. Petrop., 15, ad annos 1799-1802, 164-9. 



61 Disquisitiones Arith., art. 323; Werke, I, 1863, 391; German transl. by Maser, 377-383. 

 B2 Archiv Math. Phys., 12, 1849, 211-276. 

 63 Nouv. Ann. Math., (3), 2, 1883, 535-542. 



