398 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xii 



the relations y n+i = x n , x n+i = 2kx n +x n -i hold. To apply to tl Dul = ( l) n , 

 when D = /c 2 +l, make the substitution t n = ~ku n -\-v n ; then u n , v n satisfy the 

 initial equation with p= 1. Hence u n +i = 2ku n +u n -i ) t n +i = 2kt n +t n -i. 

 A. Cunningham 290 discussed the values of D for which 



(a&l) 2 -(a&) 2 =0 (mod (24Z)) 2 ), 

 where a, b are of the form 2Dn-\-l and Dx-y 2 = ab. 



H. B. Mathieu 291 asked if (m 2 I)x 2 +l = y~ has solutions not given by 



E. Dubouis 292 stated that there are no others in view of Fontene", 275 the 

 exposition by Legendre being insufficient. All the solutions can be found 293 

 by applying Gauss, Disq. Arith., art. 200. 



R. Fueter 294 noted that Dirichlet 108 gave sufficient, but not necessary, 

 conditions that x 2 my z = 4 be solvable for certain positive integers m not 

 squares. When m = 1 (mod 8), x and y are even and the problem reduces to 

 x 2 my 2 = 1; a necessary, but not sufficient, condition that it be solvable is 

 that in the domain defined by V m there be an even number of classes in 

 every genus. 



A. Cunningham 295 wrote r' x , v' x and T X , v x for the xth solutions of 

 T fS -2v' 2 = -1, r 2 -2*; 2 = l, and noted that E. Lucas (Ch. XVII of Vol. I of 

 this History) proved that every prime p divides some v x) where x = (p l}/n 

 when p = 8col, x = (p+l)ln when p = Sco=b3, and n = 2m. It is here 

 proved that, if n = 4m, 8m, 16m or 32m, then p = 8co+l=a 2 +6 2 = 

 with 5 = 4/3, d = 28, and the number of factors 2 of n is given. If n = 

 either p = 8wl = 3co'+l, p = (?+6# 2 , or p = 8co3 = 3co'-l, p = 2G*+3H. 



St. Bohnicek 296 proved that if TT is a semiprimary prime in the domain 

 R defined by a fourth root of unity and if the norm of TT is =1 (mod 8), 

 2 7T77 2 = 2, ? 7r77i = l have the solutions 



/772 _ '7 12 r 



so that ^, 77 are odd, ^i and 771 integral numbers in R. Here T^ 

 T^IlSizs+i, where S r is the lemniscate function defined (p. 680) in terms of 

 Jacobi's theta functions. But ^ wtf = i or 2i is not solvable in integral 

 numbers with , 77 odd in the second case. If TT is semiprimary, 2 7r?7 2 = 4, 

 1 i^h = 4i have odd solutions ^, 77 in R only if ?r = 1 (mod X 4 ) , TT ^ 1 (mod X 5 ) , 

 where X = l+i. There are similar theorems for 2 7rr; 2 = l, 2, i or 2i, 

 when the norm of TT is not = 1 (mod 8). Application is made (pp. 719-725) 

 to x 2 py 2 = 1, 2, 4, where p is a rational odd prime, use being made 

 of cyclotomic functions. 



E. E. Whitford 4 gave an extended history of Pell's equation and (pp. 

 98-112) extended the tables of Degen 101 and Bickmore 219 by listing for 



290 L'interm&liaire des math., 18, 1911, 166-7. 



291 Ibid., 220. 



292 Ibid., 19, 1912, 47. 



293 L'interm&liaire des math., 19, 1912, 72. 



294 Jahresber. d. Deutschen Math.-Vereinigung, 20, 1911, 45-46. 

 296 British Assoc. Report for 1912, 412-3. 



299 Sitzungsber. Akad. Wiss. Wien. (Math.), 121, Ha, 1912, 701-7. 



