376 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xn 



C. Hermite 129 proved that if D is positive, x 2 Dy 2 = l has an infinitude 

 of integral solutions, and all are given by 



x+y^D = (a+bJDy (i = 0, 1, 2, ) 



where a, b are solutions such that a+b^D is a minimum. 



A. Genocchi 130 proved the results stated by Jacobi 112 by means of 

 4Z [cf. Dirichlet 113 ]. For x = i= V^T, let Y and Z become 

 i and z+Zii. According as the prime p is of the form 8/c+3 or Sk+7, 

 we have 



n n 



77 - / Y\y'* t ~ 



A 4 p / 

 where the product extends over the (p 1)/2 quadratic residues of p. 



P. L. Tchebychef 131 proved that if o>, ft give the least positive solutions of 

 x~Dy 2 = l, and if x 2 Dy 2 =N is solvable, one solution has 



If a, 6 and cti, 61 are solutions a:, ?/ within these hmits of x 2 Dy 2 =N, 

 then (a&i+ai&)(a&i ai&) is a multiple of N, while neither factor is. Hence 

 if x 2 Dy 2 =N has two distinct sets of solutions within these limits, N is 

 composite. 



A. Gopel 132 proved, by use of continued fractions, that if A is a prime 

 of the form 4A'+3 or the double of such a prime, x 2 Ay 2 =2 is solv- 

 able, the sign being -f- or according as A (or %A) is =7 or 3 (mod 8), 

 and related theorems as to the values of A for which x 2 Ay 2 = 2 is solvable, 

 to be given in Vol. Ill under binary quadratic forms. 



G. L. Dirichlet 133 noted that if f = ax 2 -\-2bxy +cy 2 has for its determinant 

 D = b 2 ac a number not a square, and if a is the g.c.d. of a, 26, c, and if 



z = Ao/+/x2/', y = vx'+py r , Ap /*? = !, 



is a transformation with integral coefficients of determinant unity which 

 transforms / into itself, then 



\=(tbu)/ff, n=cu/ff, v = au/<T, p (t-\-bu)lff, 



where t, u are integral solutions of t 2 Du 2 =<? 2 ; and conversely, if t, u are 

 integral solutions, the values of A, , p are integers which determine a 

 transformation of / into itself. For the more difficult case in which D is 

 positive, and / is a reduced form, obtain from the period of reduced forms 

 defined by / all the transformations of / into itself and hence, by the above, 

 deduce all solutions of t 2 Du 2 =ff 2 . This theory, which will be given under 

 binary quadratic forms in Vol. Ill, is closely connected with the continued 

 fraction for the positive root of a+26co+cco 2 = 0. 



129 Jour, fur Math., 41, 1851, 209-211; Oeuvres, I, 185-7. 



130 M6m. Couronne"s Acad. Sc. Belgique, 25, 1851-3, IX, X. 



131 Jour, de Math., 16, 1851, 257-265; Oeuvres, I, 73-SO; Sphinx-Oedipe, 10, 1915, 4, 18. 



132 Jour, fur Math., 45, 1853, 1-14. Summary, ibid., 35, 1847, 313-8; Jour, de Math., 15, 



1850, 357-362. Cf. Smith, 139 123, p. 783; Coll. Math. Papers, I, 284-8. 



133 Abh. Akad. Wiss. Berlin, 1854, 111-4; French transl., Jour, de Math., (2), 2, 1857, 370-3; 



Werke, II, 155-8, 175-8. Zahlcnthcorie, 62, 83, 1863; ed. 2, 1871; ed. 3, 1879; ed. 

 4, 1894. Cf. H. Minkowski, Geometric der Zahlen, 1, 1896, 164-170. 



