378 HISTORY OF THE THEORY OF NUMBERS. [CHAP. XII 



G. C. Gerono 138 proved, following Lagrange, 85 that z 2 ra/ 2 = l has an 

 infinitude of integral solutions, if n is positive and not a square. If x, y 

 are positive integral solutions, x/y is a convergent of even rank of the con- 

 tinued fraction for Vrc and corresponds to the next to the last incomplete 

 quotient of one of the periods. 



H. J. S. Smith 139 stated the principal theorems relating to t 2 Du 2 = 1 or 

 4 by use of Euler's 72 notation (q i} , q n ). He noted, as had Lagrange 75 

 and Gauss 93 (Art. 222), that the methods used by Euler 65 - 71> 83 are incomplete 

 because he always assumed that a first solution is known and merely 

 deduced from it those solutions which belong to the same set, whereas there 

 may exist solutions belonging to a different set, and lastly because he gave 

 no method to distinguish between the integral and fractional values con- 

 tained in his formulas for x, y. 



L. Kronecker 140 noted that if T, U are the least solutions of T 2 -PU 2 = 1, 

 log (T+ U VP) can be expressed in terms of special theta functions or elliptic 

 functions, and the number of classes of binary quadratic forms of deter- 

 minant P. He deduced approximate values for T, U; likewise, for the least 

 solutions 4, 1 of t 2 17u 2 = 1, 4+ Vl7 has the two approximations 



o -j 



_ .,(1/10) T Js5 



R. Dedekind 141 proved the existence of integral solutions t, u ( 

 of t 2 Du 2 = l by the method used by Dirichlet 120 for complex integers, 

 but replacing his lemma by the following: There exist infinitely many pairs 

 of integers x, y such that x 2 Dy 2 is numerically <l+2 VZ). 



C. Richaud 142 stated that x 2 Ny 2 = 1 is solvable for various types of 

 values of N: If A, , L are primes of the form 8n+5 and N = 2A a , 

 2A 2a+1 2 < 3+1 or 2A**B V -L 2A . If B, - , L are not included among the 

 linear divisors of t 2 -2Au 2 , and N = 2A a B (i , 2A a B w+1 C Zy+1 or 2A Za+l B Zfi -L z \ 

 If a, b, -,l are primes of the form 8n+l, and are not included among the 

 linear d'.visors of t 2 -2Au 2 , and N = 2A 2m+1 a a , 2A 2w+1 a 2a+ V* +1 or 2A 2m+1 a Za 

 - - -Z 2A . If A, - , L are primes not included among the linear divisors of 

 J 2 -om 2 , where cois a prime 4n+l, and N = ai m A Za+1 , rf m + l A, ^ m+1 A Za+1 B^ +1 , 

 co 2m+1 A 2a ---L 2 \ Also for 8 more such sets of N's. He 143 proved these 

 results and similar ones by use of the continued fraction for VAT and the 

 reciprocity law for quadratic residues. 



Richaud 144 gave minimum integral values of x, y satisfying x 2 Ay- = l 

 for A = a?d (d a divisor >1 of 2a) and for many values of A such as 

 (9a+3) 2 9, (9a+6) 2 9, (25a+5)' 2 -25. Likewise for x 2 -Ay 2 = --1. 



138 Nouv. Ann. Math., 18, 1859, 122-5, 153-8. 



139 Report British Assoc., 1861, 96, 97, pp. 313-9; Coll. Math. Papers, I, 195-202. 



140 Monatsber. Akad. Wiss. Berlin, 1863, 44; French transl. in Annales sc. de 1'ecole normale 



sup., 3, 1866, 302-8. Cf. Smith, Report British Assoc., 1865, 138, p. 372; Coll. Math. 

 Papers, I, 354-8. 



141 Dirichlet's Zahlcnthcorie, 141-2, 1863; ed. 2, 1871; ed. 3, 1879; ed. 4, 1894. 



142 Jour, de Math., (2), 9, 1864, 384-8. 



143 Ibid., (2), 10, 1865, 235-280; (2), 11, 1866, 145-176. 



144 Atti Accad. Pont. Nuovi Lincei, 19, 1866, 177-182. 



