408 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xm 



D. Hilbert 54 remarked that the proof that a proposed diophantine 

 equation is not solvable in rational numbers is often made by showing that 

 the corresponding congruence with respect to a prime or prime power 

 modulus is impossible. For the case of a quadratic equation in two variables 

 it follows conversely that the possibility of solving the congruence for 

 every prime power modulus implies the possibility of solving the equation. 

 For, the known cr terion for the solvability of a ternary quadratic dio- 

 phantine equation leads to the result : If m, n are any integers, the equation 

 mx 2 -\-ny 2 = l is solvable for rational numbers x, y, if the congruence 

 mx 2 -\-ny 2 =l (mod p e ) is solvable in integers x, y for every prime p and 

 positive integer e. There is no immediate extension to higher equations, 

 since 



is irreducible and has no rational solution, while the corresponding con- 

 gruence modulo p e is solvable whatever be the prime p and positive integer 

 e. Again, Z 4 +13 2 +81 is an irreducible function which becomes reducible 

 modulo p e for every prime p and integer e. 



Several writers 55 found all solutions of x(x-\-l}/2 = y(y+l}/3 by means 

 of 2u 2 -3z 2 =-l. 



On Mx 2 Ny 2 =l or 4, see Legendre, 88 Jacobi, 112 Weber, 218 Palm- 

 strom, 228 and de Jonquieres 235 of Ch. XII. 



On x 2 -\-qy 2 = m see Cornacchia 4 of Ch. XXIII. 



On ax 2J rcy 2 = n, see Euler 56 and Nasimoff. 68 



SOLUTION OF ax 2 +bxy+cy 2 = k. 



L. Euler 50 noted that the problem to find the minimum of Ax~+2Bxy 

 +Cy 2 for integral values 4=0 of x, y presents no difficulty if B 2 AC^O 

 and hence is here treated for B 2 AC positive and not a square. Then 

 the proposed form may be reduced to mx 2 ny 2 , where m and n are positive 

 integers whose ratio is not a square. If m=l, it can be given the value 

 unity by Pell's theorem. If n = l, it can be given the value 1. 



If mx 2 ny- = k for x = a, y = b, it has an infinitude of solutions. For, 

 if p 2 mnq z = l (in an infinitude of ways, since mn^ D), then 



mx 2 ny 2 = (ma 2 rib 2 ) (p 2 mnq 2 ) \ 

 This holds if 



x 



so that we get x, y as rational functions of a, b, p, q. 



The problem to make mx 2 ny 2 a minimum corresponds to finding the 

 rational fraction x/y giving the closest approximation to Vn/ra. Develop 

 the latter into a periodic continued fraction and take the convergent ob- 

 tained by continuing to the largest quotient. Thus, for 7x 2 13?/ 2 , the con- 



M Gottingen Nachrichten (Math.), 1897, 52-54. 

 66 L'interme'diaire des math., 22, 1915, 239, 255-260. 



66 Novi Comm. Acad. Petrop., 18, 1773, 218; Comm. Arith. I, 570; Opera Omnia, (1), III, 

 310. On the incompleteness of Euler's methods, see Smith 139 of Ch. XII. 



