564 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxi 



whence s 3 = 6wri 2 . Take i = 3n 3 , 7* = 4u 2 ra 3 , whence s = Qumn 2 . Hence solu- 

 tions are a\, a 2 = 4i* 3 m 3 3ft 3 . Next, for a 3 = p s, our initial equation 



i 



becomes 



3ps 2 / , ps 2 \ 3 3w 3 s 



if = -- 



n n n \ nr 



Special sets of five cubes whose sum is a cube have been noted. 117 

 A. Martin 118 noted that the sum of the cubes of p+q, p q, r p, s is 

 the cube of r-\-p if p = ^s 3 /(r- q*'); that of a+b c, a+c b, b-\-c a, y is 

 the cube of a+6+c if y 3 = 24abc, whence take a = 3p 3 , 6 = 3g 3 , c = r 3 or take 

 y = 2a, c = a 2 /(36); the sum of the cubes of pa-\-nt, qant, ra nt, nt is of 

 the form sa?-\-R and is a cube if s^p 3 +^ 3 +r 3 is a cube and if R = Q, which 

 determines t. Next, he gave Whitley's 109 result. 



Finally, given that p{-\ ----- \-pl is a cube, to find n+1 cubes whose sum 

 is a cube. If n is odd, take x, p\x, p^x, pz+x, px, p$+x, , p n +x 

 as the roots of the desired cubes, where 



If n is even, take x, pi+x, p^x, p^+x, px, , p n -\-\-x, p n x as the 

 roots, and (+#) 3 as the sum of their cubes, where 



Martin 119 found cubes whose sum is a cube b 3 by selecting 6 3 between 

 n z and AS = 1 3 +- -+n 3 and seeking by trial to express S b 3 as a sum of 

 distinct cubes ^ n z . Also by seeking to express p s q 3 as a sum of distinct 

 cubes 4=5 3 . He tabulated the values of S for n si 342. 



R. W. D. Christie 120 gave 14 cases like 4 3 = l + l+2 3 +3 3 +3 3 of a cube 

 equal to a sum of five cubes. 



Ed. Collignon 121 noted that there is no positive integral solution of 



x*+(x-l) p -\ ----- k-(x-k)v = (x+l)v-\ ----- \-(x+k) p (p = 3or4). 



A. Gerardin 122 gave numerical examples of equal sums of three cubes. 

 A. S. Werebrusow 123 noted that (1) holds if 



x = u-\-v, y = u v, u = a?m*, v = bn 3 , z=6mri 2 , ab = Q. 



From two sets of solutions a third set is derived. 



A. Gerardin 124 gave, besides two more complicated identities of like type, 



Gerardin 125 discussed a 3 +6 3 +/ic 3 = (a+6) 3 + /id 3 . For a = pm, c = 



117 Amer. Math. Monthly, 2, 1895, 329-331. 



118 Math. Magazine, 2, 1895, 156-160. 



119 Ibid., 185-190. Two examples, Martin 68 of Ch. XXIII. 



120 Math. Quest. Educ. Times, (2), 4, 1903, 71. 



121 Sphinx-Oedipe, 1906-7, 129-133. 



122 Ibid., 120-4. 



111 Math. Soc. Moscow, 26, 1908, 622-4. 

 12 Assoc. fran<j., 38, 1909, 143-5. 

 126 Sphinx-Oedipe, 5, 1910, 178. 



