364 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xii 



Euler 84 solved ax 2 + 1 = ?/ 2 for special types of numbers a. Given 

 p 2 = 5 2 -f- c 2 , determine g, f so that bg cf = 1 and take q = bf + eg, 

 a =/2_|_02. then ap 2 -l=g 2 , x = 2pq, y = 2q*+l. Next, if ap 2 =F2 = g 2 , the 

 divisor p 2 of <z 2 d=2 must be of the form 6 2 2c 2 ; hence take a=/ 2 2<7 2 , 

 cfbg = l or 1, q = bf2cg. If ap 2 =b4 = g 2 , the divisor p 2 of g 2z F4 must be 

 of the form 6 2= Fc 2 ; hence take a=/ 2 T# 2 , cfbg = 2 or 2, q = bf : fcg. 



Lagrange 85 simplified his 7a method applicable to equations of any degree. 

 Of two methods to solve F=Cy 2 2nyz-\-Bz 2 = l in integers, one is to 

 render F a minimum, and the other consists in applying transformations 

 which replace F=l by L, Z 2N &-}-M$~ = 1, where 2 N exceeds neither 

 | L nor | M , while the determinants N 2 LM and n 2 (75 = A are equal. 

 By multiplication by M , we get v 1 A = M where v M\fr N%. If A = a, 

 where a > 0, it is proved that = 0, M = 1 . If A > 0, v/t- is a convergent of 

 the continued fraction for VA. Euler's 77 example, 101 =x 2 13?/ 2 is now 

 (pp. 614-620) transformed into z~ 13w 2 = 1 which is solved by use of the 

 continued fraction for Vl3. 



Euler 108 of Ch. XXII deduced an infinitude of solutions of a 2 -A/3 2 = 4 

 from one solution. 



Petri Paoli 86 treated a+c 2 x 2 = y z . Since a is a difference of two squares, 

 set y = cx-\-l, cx+2, , in turn. Then a = 2cx-\-l, 4cx+4, 6c+9, 

 For a odd, use the first, third, terms, so that x will be an integer chosen 

 from the series (a l)/(2c), (a 9)/(6c), . Similarly for a even. If a is 

 positive, the terms of the series decrease and there is a finite number of 

 trials. The case in which a is negative can be reduced to the preceding. 



A. M. Legendre 87 obtained important conditions for the solvability of 

 equations of degree 2 by use of Lagrange's 75 method for x-By 2 = A, 

 where A and B are integers with no square factor and A>B>0. By that 

 method, 



(15) c?-B = AA'&,a'*-B = A'A"k", ,ci^A/2,a' = A'a^A'/2, , 



where A', have no square factors, and A M < B, so that the proposed 

 equation depends upon 



(16) x*-By* = A', x*-Bif = A.", -, .x 2 - By* = A< n \ 



Legendre proved that, if for x z By z = A and the first transformed equation 

 (16) there exist integers a, a', /?, 0' such that 



a^B (mod A}, a' 2 =B (mod A f ), p=A, /3' 2 =A' (mod 5), 



the like conditions hold for the second transformed equation (16). Since 

 a" 2 =B (mod A"), by (15), it remains only to prove the existence of an 

 integer 0" for which j8 //2 =JL" (mod B). If is a prime factor of B, we 



84 Opusc. Anal., 1, 1783 (1773), 310; Comm. Arith. Coll., II, 35-43. 



85 Additions to Euler's Algebra, Lyon, 2, 1774, pp. 464-516, 561-635; Oeuvres cle Lagrange, 



VII, 57-89, 118-164; Euler's Opera Omnia, (1), I, 548-573, 598-637. 



88 Opuscula analyttaa, Liburni, 178,0, 122. 



w M6m. Acad. Sc. Paris, 1785, 507-513. Cf. Legendre, Thdorie des nombres, 1798, 43-50; 

 ed. 2, 1808, 35-41; ed. 3, 1, 1830, 41-48; German transl. by Maser, I, 41-49. In his 

 texts, Legendre introduced the factor z- in the right members of (16). 



