CHAP, xxi] NUMBERS THE SUM OF Two RATIONAL CUBES. 577 



and found an infinity of solutions from one by treating 



1+x 3 (mx+ri) 3 =(l m 3 )(x a)(x &)(x 7) 



by his method 238 for x 3 +y 3 = z 2 to obtain (7 3 +l) J and 7 as functions of 



= /?. 



A. S. Werebrusow 213 discussed the form of numbers A expressible as the 

 sum of two rational cubes. Elsewhere he 214 took 



x+y = Aozl, x*-xy+y- = AiZ*, A=A Ai, z 

 whence A i is of the form (s, t) = s 2 +st+t' 2 , and z\ = (a, 6). Then 

 2? = (M, N), M = a 3 +3a 2 6-6 3 , N = -a 3 +3a6 2 



Aizl = (s, t)(M, N), x=(s+t)M+sN, y = tM+(s+)N, 



with similar formulas derived by interchanging s and t or M and N. Further 



treatment was given for z\ = 1, Ai = l, 3 or 7. 



A. Cunningham 215 discussed x 3 y 3 = 17z 3 , obtaining integral solutions 



with z = 7. From the solution x = 18, y = 1, z = 7 of x 3 +y 3 = 17z 3 , Prestet's 



formula leads to positive integral solutions smaller than those given by 



Lucas' (1). 



R. W. D. Christie 216 noted results due to Desboves. 203 



Christie 217 noted that, if p = a 3 -6aZ> 2 -3a 2 6-& 3 , X 3 -pY 3 = l has the 



solution 



_ = 



' y ~ 



3a6(a+6) ' ~3ab(a+b) ' 



and hence also X = l/x, Y= y/x. 



A. Cunningham 218 treated x*+y 3 = Cz* for x, y relatively prime by setting 



x+y = X, x^-xy+y^Y, z={Z. 



The g. c. d. of X,Ylsl or 3. Let C be prime to 3. Then XY = C?Z 3 , 

 X = C{ 3 , Y = Z 3 - or f = 3f, Z = 9Cr /3 , Y = 3Z 3 . 



Since Z 3 is a factor of Y and is prime to 3, Z = A 2 +3B 2 . Hence Z 5 = A\ +3B?. 

 But, for y even, F=(z-|?/) 2 +3(^/) 2 . Hence, if Y = Z 3 , x-%y=A lt 

 If Y = ZZ*,x-%y = 3B lt ^y = A^ For y odd, 



There is treated also the case C=0 (mod 3). 



T. Hayashi 219 concluded from the impossibility of rational solutions of 

 x 3 +y 3 = 3z 3 that 4a(a+/3)(a+2/3)/6 is never a cube. 



R. D. Carmichael 220 noted that, if A = 2 m , we may take x, y, z odd and 

 proved that one of the variables must be zero, except for the trivial solution 

 x = y = z which occurs if m = l. 



213 Matem. Shorn. (Math. Soc. Moscow), 23, 1902, 761-3. 



214 L'intermediaire des math., 9, 1902, 300-3. 



216 Math. Quest. Educ. Times, (2), 2, 1902, 38 [48], 73. 

 IUd., (2), 3, 1903, 109-110. 



217 Ibid., (2), 13, 1908, 90. Cf. Desboves. 203 



218 Ibid., 27-30. 



219 Nouv. Ann. Math., (4), 10, 1910, 83-6. 



220 Diophantine Analysis, 1915, 70-72. 

 38 



