CHAP. XX] BINARY QUADRATIC FORM MADE AN ?ITH POWER. 539 



A. Gerardin 57 summarized the known results on x 3 k = z 2 ; he noted 

 the solutions 2 3 4 = 2 2 , 5 3 4 = IP, contrary to de Jonquieres' 13 assertion 

 that only one solution exists. Given one solution x 0) z , Gerardin deduced 

 (ibid., 163-5) the second solution [Realis 17 ] 



Set x Q = 2p, where p is a prime. Then z = p y , 2p j , 3p y , 6p j (j = 0, 1, 2). 

 There result twelve integral values of k for which the given equation is 

 solvable. For &=(2p-l) 2 (9p 2 -2p + l), the solutions include x = 2p, 

 2p-l, 2-4p, 4p 2 -2p, (12p 2 -6p-hl) 2 -4p+l. 



L. Bastien 58 listed the values 3, 5, 6, 9, 10, 12, 14, 16, 17, , 99 of 

 n^lOO for which q 3 k 2 = n is impossible, the values n = 1, 2, 8, 13, 29, , 

 81 for which there is a single solution, and the values for which there are 

 more than one solution. 



Crussol 59 noted cases when x 3 -\-k = y 2 has 7, 9, 34 and 41 solutions. 

 A. Gerardin (ibid., p. 16) noted cases when it has 21 solutions. 



A. Gerardin 60 proved that all solutions of x 2 +3y 2 = z 3 are given by 



(a 3 -9a/3 2 ) 2 +3(3a 2 /3-3/3 3 ) 2 =(a 2 +3/3 2 ) 3 . 



T. Hayashi 61 proved that y*+l*s& for 7/ + 0; 7/ 2 -l=M 3 for i/ 2 + 0, 1, 9. 



A. Cunningham 62 proved that, if p is prime, z 3 p 2 = 2-10 6 has the single 

 solutions = 129, p = 383. 



L. J. Mordell 62a noted that no equation x 2 +a = y 3 has an infinitude of 

 integral solutions. 



For 2z 2 +2z+13 = 2/ 3 , see paper 161 of Ch. I. On 276 2 +l = 4c 3 , see 

 Kronecker 23 of Ch. XXI. 



BINARY QUADRATIC FORM MADE AN HTH POWER. 



J. L. Lagrange 63 noted that the mth power of f=x 2 +axy-\-by 2 can be 

 expressed in the same form F = X 2 +aXY+bY 2 by employing the factors 

 x-\-ay, x-\-py of /and taking X-\-aY to be the expansion of (x-\-ay) m . The 

 resulting values of X, Y make F an mth power. 



L. Euler 64 stated that he used this method for /=x 2 +ra/ 2 in the first 

 edition of his algebra. 6 



Euler 65 noted that, if N = a?+nb 2 , N x is of the form x 2 +ny 2 , and asked 

 for the least z+0 or least ?/=|=0 for which N* = x 2 -\-ny 2 . Let 



67 Sphinx-Oedipe, 8, 1913, 145-9. 

 58 Ibid., 9, 1914, 15-16. 

 69 Ibid., 43-44. 



60 L'intermediaire des math., 21, 1914, 129. 



61 Nouv. Ann. Math., (4), 16, 1916, 150-5. 



62 Math. Quest, and Solutions (3), 3, 1917, 74. 



620 London Math. Soc. Records of Proceedings, Nov. 14, 1918. 



63 Addition IX to Eider's Algebra, Lyon, 2, 1774, 636-644; Euler's Opera Omnia, (1), 1, 1911, 



638-643; Oeuvres de Lagrange, VII, 164-170. For f=x z -By 2 , Lagrange, Mem. Acad. 

 Sc. Berlin, 23, annee 1767; Oeuvres, II, 522^. 



64 Opera postuma, I, 1862, 571-3, letter to Lagrange, Jan., 1770; Oeuvres de Lagrange, 



XIV, 216. 

 M Nova Acta Acad. Petrop., 9, 1791 (1777), 3; Comm. Arith., II, 174-182. 



