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



to 2D, such that for all primes p and q satisfying 



p=%, q=ri (mod 8/S>Di), 



the equation PpqDu? = l has a fundamental solution T, U for which 

 neither T+l nor T 1 is divisible by pq. 



L. Lorenz 151 found the number of integral solutions of m 2 +en 2 = ]V, 

 where e 1, 2, 3, 4 or 1, and N isa given positive integer, by transforming 

 the series 



TO, n= oo 



N 



into a series of another form and finding the term q N of the latter. For 

 details when e = 1 see Lorenz 94 of Ch. VI. 



P. Seeling 152 noted jthat, if A is positive and not a square, and the con- 

 tinued fraction for VA has the symmetric period n; a, b, - -, b, a, 2n, 

 solutions x, y of a; 2 Ay 2 = d=l are given by the numerator and denominator 

 of the convergent belonging to the quotient In. The sign is plus if the 

 number of quotients in the period is even; while if it be odd, the sign is 

 plus after 2, 4, 6, , periods, minus after 1, 3, 5, periods. If 

 x 2 Ay-= 1 and the number of quotients in the period is odd, then 

 A = 4w+l or 4ra+2 and A has no factor 4ra+3; if A is a prime 4m+l, the 

 number of terms in the period for VZ is odd ; if A is a product of two or 

 more primes 4m+ 1 or the double of such a product, no general rule has been 

 found. Finally, he tabulated all numbers A < 7000 for which the per'od of 

 VZ has an odd number of terms, so that x L Ay i = 1 is solvable. 



A. B. Evans and A. Martin 153 found the least solution of rx 2 -}-l = \3, 

 where r = 940751, and noted that rx 2 +38= D has no integral solution. 



Moret-Blanc 154 noted that if x = h, y = k is a solution of 2x* 1 = 7/ 2 , then 

 x = hu+kv, y = 2hv j rku give a second solution, provided u 1 2y 2 = l, as for 

 u = 3 } v = 2. 



F. Didon stated and C. Moreau 155 proved that, if D= (4n+2) 2 +l, where 

 n is a positive integer, t z Du~ = 4: has no solution in odd integers, and the 

 least positive solution is i = 16(2n+l) 2 +2, w = 8(2n+l). 



0. Schlomilch 156 discussed the continued fraction for Va 2 /4/3. 



L. Matthiessen 157 noted that if =/, y = g give the least solution of 

 ax 2 2/ 2 = l, all solutions are given by 





161 Tidsskrift for Math., (3), 1, 1871, 97. Cf. *J. Petersen, ibid., p. 76. 



152 Archiv Math. Phys., 52, 1871, 40-9. 



153 Math. Quest. Educ. Times, 16, 1871, 34-6. 

 164 Nouv. Ann. Math., (2), 11, 1872, 173-7. 

 m lbid., 48; (2), 12, 1873, 330-1. 



166 Zeitschrift Math. Phys., 17, 1872, 70-71. 

 157 Ibid., 18, 1873, 426. 



