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



proved that, when A = 2a, where a is a prime, 2P au? = +1 is solvable for 

 a = 8n+7, 2t 2 -au?=-I for a = 8n+3, and t~-2au 2 =-l for a = Sn+5. 

 This method of exclusion yields no result when a = 8w+l. But using 

 also the quadratic reciprocity law, he proved that P2au-= 1 is solvable 

 if a is a prime 16n+9 such that 2 (a ~ 1) / 4 = - 1 (mod a), though the conditions 

 are not necessary. If a and b are both primes 4n+3, at 2 bu?=(a/b) is 

 solvable.* If a and b are both primes 4n + 1 and if (a/b) = l,t 2 abv? = 1 

 is solvable; but if (a/6) = 1, and* (a/6) 4 =-l, (6/a) 4 =-M 2 -a6tt 2 =-l is 

 solvable, though the conditions are not necessary. He gave criteria for 

 the solvability of t 2 abcu z = I, where a, b, c are primes 4n-f-l. Finally, 

 Dirichlet removed the initial hypothesis that p, q give the least solution of 



M. A. Stern 109 developed the theory of continued fractions and in the 

 final article (pp. 327-341) made application to x 2 Ay* = D, in particular 

 when D = l, 2. He tabulated 42 forms for A, like m 2 n 2 +2ra and 

 (6nd=l) 2 +(8n2) 2 , such that there is a small number of explicitly given 

 partial denominators in the continued fraction for VA, whence one finds 

 at once the least solution of x 2 Ay* = l. 



B. Peirce and T. Strong 110 solved 376z 2 +114z+34 = 7/ 2 by setting 

 376z +57 = a/ and treating 376?/ 2 -z' 2 = 9535 by the theory of binary quad- 



T*f} tlP T OT*TY1 S 



C. Gill 111 noted the solution (1364557) 2 -369(71036) 2 = 25 and that in 

 the least solution of P 940751w 2 = l, u has 55 digits and t has 58 digits. 



C. G. J. Jacobi 112 stated that, if p is a prime 4n+l, and x i py' i = 4, 

 then 



Jp (x+y 4p) = 2<p +1 > / 2 n sin 2 , 



where a ranges over the quadratic residues, between and p/2, of p. If q 

 is a prime 8n-f 3, and x z qy*= 2, then 



If q and q' are primes 4w+3, and q is a quadratic residue of q', then 



fq'y, 



q q 

 where gz 2 gV = 4. Cubing f ( V#c+ Vg ; 2/), we get solutions of qu z q'i ff = 



* Legendre's symbol (0/6) denotes + 1 or 1 according as x z = a (mod 6) is solvable or not. 



Let c be a prime 4n + 1, and fc an integer not divisible by c for which (/c/c) = + 1, 



viz., fc''" 1 '/ 2 s + 1 (mod c). According as fc (c ~ 1)/4 = + 1 or 1 (mod c), Dirichlet wrote 



(fc/c) 4 = + 1 or 1, respectively. 

 "Uour. fur Math., 10, 1833, 1-22, 154-166, 241-274, 364-376; 11, 1834, 33-66, 142-168, 



277306 311350 



110 Math. Miscellany, 1, 1836, 362-5; French transl., Sphinx-Oedipe, 8, 1913, 117-9. 



111 Ibid., 230. 



112 Monatsber. Akad. Wiss. Berlin, 1837, 127; Jour, fur Math., 30, 1845, 166; Werke, VI, 



263-4; Opuscula Mathematica, 1, 1846, 324-5. Proof by Genocchi. 130 



