CHAPTER XXIII. 



EQUATIONS OF DEGREE n. 



SOLUTION OF /= CONST., WHERE / is A BINARY FORM. 

 J. L. Lagrange 1 noted that, in seeking integral solutions of 



A =Bt n +Ct n ~ l u-{ ----- \-Ku n , 



where A, , K are given integers, we may take u relatively prime to A, 

 and thus find integers 0, y such that t = udAy. Inserting the value of t, 

 we see that 50 n +C0 w ~H ----- \-K must be divisible by A. If such an 

 integer 6 exists, the proposed equation reduces, after division by A, to 



F(u, y)^Pu n +Qu n - l y+..- + Vy n = l, 



where P, , V are given integers. Set ufy = x, F(x, 1) =z. Then l/y n = z. 

 The problem of solving F = 1 in integers reduces to the examination of the 

 real values a of x for which z is zero or a minimum (whence dz/dx = 0}. 

 For such an a, Lagrange employed the continued fraction for a and two 

 series of convergents and proved that u/y must equal one of these conver- 

 gents IjL, whence w=dbZ, y = L. While a root of z = may lead to an 

 infinitude of solutions, a root of dzfdx = furnishes only a limited number. 



A. M. Legendre 2 reproduced this method of Lagrange's, developing 

 into a continued fraction each real root of F(x, 1) =0 and also the real part 

 of each imaginary root and forming their various convergents p/q. The 

 least of the F(p, q) is the minimum of F(u, y) for integral values u, y. In 

 case the minimum is 1, we have a solution of F(u, y) = l and hence a 

 solution of the initial equation A = Bt n + 



H. Poincare 3 noted that the problem reduces to the case of the repre- 

 sentation of a number N by a form in which the leading coefficient is unity : 

 x m -\-Ax m ~ 1 y+ . We first solve the congruence m A% m ~ l -{- - =0 (mod 

 N) and then determine by Hermite's method whether or not two decompos- 

 able forms in m variables are equivalent under m-ary linear transformation. 



G. Cornacchia 4 gave a method of solving in integers 



(1) 



A=0 



when Co and C n are positive and a root x Q >P/2 of the corresponding con- 

 gruence "2ChX n ~ h =0 (mod P) is known. Take y such that x y = l 

 (mod P). Apply the g.c.d. process to P, x 0) and let xi, x 2 , -, x m = 1 be the 

 remainders. Let yi, - , y m = 1 be the corresponding remainders from P, y . 

 Then if (1) has relatively prime integral solutions a, b such that 2ab<P, 

 this solution is one of the above pairs x^, y m +\-i or is a pair obtained 



1 Mem. Acad. Berlin, 24, annee 1768, 1770, 236; Oeuvres, II, 662, 675. For n =2, Lagrange 76 



of Ch. XII. 



2 Theorie des nombres, 1798, 169-180; ed. 3, 1830, I, 179; German transl., Maser, I, 179. 

 8 Comptes Rendus Paris, 92, 1881, 777. Cf. Poincare. 24 



4 Giornale di Mat., 46, 1908, 33-90. 



44 673 



