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



A. Auric 268 developed into a continued fraction the root of any quadratic 

 equation of discriminant A; it is a question of factoring 2, where t, u 

 give the least solution of t 2 Aw 2 = 4. 



B. Niewenglowski 269 noted that x 2 cw/ 2 = 1 is solvable if and only 

 if the least positive integral solutions of x 2 m/ 2 =-|-l are of the form 

 x = l-{-2u 2 , y = 2uv. The latter represents an hyperbola; if P and P\ are 

 points on it with integral coordinates, the line through P parallel to the 

 tangent at PI cuts the hyperbola in a new point with integral coordinates. 



A. Cunningham 270 gave tests for the divisibility of solutions of 



by a prime. 



The existence of a fundamental solution of Pell's equation is a corollary 

 to Dirichlet's theorem on the units in any algebraic field. For the case of 

 a quadratic field, reference may be made to J. Sommer's 271 text. 



" E. A. Majol " 272 gave eight values, 75, 78, 321, , of A for which there 

 is a common prime divisor 4m +3 of A and y in the fundamental solution of 

 x z Ay z =l. 



A. Boutin 273 gave the period of the continued fraction for VA for many 

 forms of A, chiefly quadratic functions of a, and for various such A's 

 listed the least solutions of x 2 Ay*=l. He listed the values of N, 

 0<N<1023, for which x z Ny 2 = 1 is solvable, a necessary and sufficient 

 condition for which is that there be an odd number of terms in the period 

 of incomplete quotients in the development of V./V. 



*C. Stormer 274 gave a simple proof of his 230 theorem and applied it to 

 solve the following problem: Given the primes pi, , p n , find all positive 

 integers N for which N(N+li) is divisible by no prime other than p lt -,p n 

 when h = 1 or 2. This is solved by the theorem that, if a = 1 or 4, all posi- 

 tive integral solutions a; of x 1 1 = ap\ l p z n n occur among the fundamental 

 solutions of the equations x z Djy z =l(i=l, , *>), where DI, , D v 

 are all the values of ap\ l - - -p e n n when ei, -, e n take independently the 

 values 1, 2. 



G. Fontene 275 proved that, if a, b give the least positive solutions of 

 x 2 ky*=l, all solutions are given by x-\-y^k = (a-\-b^k) n ; the proof is 

 essentially the classic proof, but follows the proof by Mile. J. Borry (ibid., 

 13, 1907, 316). 



A. Chatelet 276 proved by an elementary formulation of the classic method 

 of solution by continued fractions that, if k is not a square, x z ky- = l is 

 always solvable. 



268 Bull. Soc. Math, de France, 35, 1907, 121-5. 



Ibid., 126-131; Wiadomosci Mat. Warsaw, 12, 1908, 1-26 (Polish). 



270 Report British Assoc. for 1907, 462-3. Cf. Cunningham. 281 



271 Vorlesungen iiber Zahlentheorie, 1907, 98-107, 113, 338-45, 355-8; French transl. of 



revised text by A. Ldvy, 1911, 103-113, 119, 351-7, 370-3. 



272 L'interme'diaire des math., 15, 1908, 142-3. 



273 Assoc. franc, av. sc., 37, 1908, 18-26. 



274 Nyt Tidsskrift for Mat., 19, B, 1908, 1-7; Fortschritle dcr Math., 39, 1908, 246. 



275 Bull. math. 616mentaires, 14, 1908-9, 209-212. 



276 Ibid., 307-331. 



