542 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xx 



A solution involving an arbitrary positive integer a is 

 x = a(f)(2a, k a 2 , ri)-\-(k a 2 )(f>(2a, k a 2 , n l), y = 4>(2a-, k a?, ri), 



z = (k a 2 ) for n even; z = (/b a 2 ), a> Vfc, for n odd. 



M. Weill 74 repeated Euler's 6 method for ax 2 +cy 2 = z n . 



T. Pepin 75 proved that, if the number of classes of quadratic forms of 

 determinant c is relatively prime to n, all relatively prime integral solu- 

 tions of x 2 +cy 2 z n are given by 



For n = 3, the solvability depends upon whether or not the triplication of a 

 quadratic form gives the principal class. 



H. S. Vandiver 76 found an infinitude of, but not all, solutions of 



G. Candido 77 employed Lucas' functions U n , V n : 



Change q to p 2 q- Thus x 2 qy 2 = z n has the solution x = %V n , y=U n , 



A. Cunningham 78 noted that y*+y + l=x n is impossible if n>3, 

 z"<2-10 8 . R. W. D. Christie stated that x must be of the form a 2 +a+l 

 and inferred that TO =4, 5, n + 3 unless a = 2. 



Cunningham 79 noted that the only solution of |(<? 2 +1) = P 4 with 

 g<lGOOOOO is # = 239, p = 13. Christie obtained this solution by making 

 special assumptions. Cf. Stormer 137 " 9 of Ch. VI; Euler 7 of Ch. XIV; Euler 53 

 and Pepin 58 of Ch. XXII. 



U. Bini 80 stated that the method of Desboves 142 of Ch. XXIII [cf. 

 Lagrange 63 ] does not lead to the determination of the form of the solutions 

 of x 2 -\-axy -\-by 2 = z n for every integer n. 



A. S. Werebrusow 81 gave polynomials X, Y of degree n in x, y making 



AX 2 +2BXY+CY 2 = (ax 2 +2bxy+cy*-) n . 

 E. B. Escott 82 noted that solutions of X 2 -DY 2 = 4:Z n are given by 



a, = 



But not all solutions are so obtained. 83 



O. Degel 84 treated the homogeneous equation obtained from the last 

 one by replacing X, Y, Z by Xifx* (i 1, 2, 3). The section C by z 2 = 



74 Nouv. Ann. Math., (3), 4, 1885, 189. 



76 Mem. Accad. Pont. Nuovi Lincei, 8, 1892, 41-72. 



76 Amer. Math. Monthly, 9, 1902, 112. 



77 Giornale di Mat., 43, 1905, 93-6. Cf. Candido. 87 



78 Math. Quest. Educ. Times, (2), 8, 1905, 09-70. 

 IUd., 9, 1906, 23-24; 14, 1908, 77. 



80 L'interm6diaire des math., 14, 1907, 246. 



81 Ibid., 15, 1908, 153; Mat. Sbornik, 22. 



82 L'intermikliaire des math., 15, 1908, 153. 



83 Ibid., 17, 1910, 2, 137-8, 229-30. 

 M Ibid., 17, 1910, 253-5. 



