692 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxm 



we get Q = A+Bp-\-Cq+Dp 2 -\ ---- . Replacing p by qy, we see that A 

 must be divisible by q. Hence we take for q the various factors of A in 

 turn and solve q = d L -\-b l x+' - for rational x's. A special treatment is 

 necessary when q reduces to the constant a 1 . G. Libri 144 eliminated x 

 between the congruences p = 0, #=0 (mod <?) and obtained D = (mod q), 

 where D is a function of the coefficients of p, q. Next, seek the integral 

 solutions of q = d for each divisor d of D in turn, and then solve y = p/q. 

 As another method he suggested (p. 317) the use of series. 

 A. J. Lexell 1440 found values of p, q, r, s for which 



pqrs (p 2 s 2 ) (q 2 r 2 ) 

 L. Euler 145 treated vW+Az 2 i/V= D, where 



r = ax 2 +2bxy-\-cy 2 , s = 

 To make s have the factor r, set 



z = agx+(f+bg)y, v=(f-bg)x-cgy. 

 Then s/r =/ 2 + (ac b 2 )g 2 = t. The proposed equation becomes 



which is of type (2) of Euler 143 , Ch. XXII. The case 6 = was treated in 

 more detail. 



G. Libri 146 treated a n x n -\-bx n ~ l -{- - - - -\-q=z n with all coefficients positive. 

 Set z = ax-\-e, whence x n ~ l (na n ~ l e 6)H ----- \-(e n <?)=0. Seek the least e 

 for which all the coefficients are positive and the greatest e for which they 

 are all negative. For each integer e within these limits, seek the positive 

 integral solutions x. If the coefficients in the given equation are not all 

 positive, set x = A-}-y and choose A so that the coefficients of the resulting 

 equation will all be positive. 



Libri 147 investigated the integral solutions ^0 of <(#, ?/, )=0 for 

 which x<a, y<b, , where a, b, are given positive integers. Set 



X = x(x-l)(x-2) . .(s-a+1), Y = y(y-l} . .(y-ft+l), . . .. 



Let F = be the result of eliminating x, y, between = 0, X = Q, Y=Q, 

 . If the equation of condition F = is satisfied, take the equation, say 

 Xi(x] =0, in one variable, preceding the final stage of elimination. Then if 

 X 2 is the g.c.d. of X\ and X, all possible integral values of x occur among the 

 roots of X z = 0; similarly for the other variables. The same method applies 

 to a congruence = (mod a) . For a a prime p,X=x p x (mod p),Y=y p y 

 (mod p). Since 



. 1 



Ls I cos - \-i sin I = 1 or 0, 

 m m / 



144 Jour, fur Math., 9, 1832, 74-75. 



144a Euler's Opera postuma, 1, 1862, 487-90 (about 1766). 



146 M6m. Acad. Sc. St. Petersb., 9, 1819 [1780], 14; Comm. Arith., II, 414. 



146 Memoria sopra la teoria dei nuraeri, Firenze, 1820, 24 pp. 



147 M6morie sur la thSorie des nombres, M6m. divers Savants Acad. Sc. de 1'Inatitut de France 



(Math. Phys.), 5, 1838 (presented 1825), 1-75. 



