CHAP. XXI] SUM OF CUBES OF NUMBERS A CUBE. 583 



for p = l+r, 



1 _p t m?--l f 

 4s 2 ~ m 2 -! I 



3mV 



2(m 2 -l) 



if r = f (4-w 2 )/(?w 3 -l), whence s=4(ra 2 -l) 2 /{18m 2 +9--(m 2 ~l) 2 }. 

 V. A. Lebesgue 249 stated that, if x and r are positive integers, 



(2) x*+(x+rY+(x+2ry i -l ---- +O+(?i-l)r] 3 =0+nr) 3 

 is impossible except for n = 3, a; = 3r. If we write 



(3) s = 2aH-(w-l)r, (r = s 2 +(n 2 -l)r 2 , 



we obtain for the left member of (2) the expression ns<r/8. He considered 

 it a difficult problem to make the latter a cube, and remarked that it was 

 impossible for n = 2 by Euler's 3 theorem. 



A. Genocchi 250 treated the last problem nsar/8 = y 3 . Set s = rt, 2y = rz. 

 Then nt(t 2 +ri*-l)=z z . Following Fermat's method, 143 set t = l+u, 

 z = n+pu, and equate the terms of the first degree in u. Hence 



n 2 +2 3n(l-p 2 ) 



(4) P=^> > u=z -r ~- 



3n p 3 n 



The cases n = 3, r = 107 ; n = 4, r = 1 ; n = 5, r = 13, give respectively 



(5) 149 3 +256 3 +363 3 = 408 3 , 1 1 3 + 12 3 + 13 3 + 14 3 = 20 3 , 



(6) 230 3 +243 3 +256 3 +269 3 +282 3 = 440 3 . 



B. Boncompagni 251 proposed for solution the same problem (2) and 



(7) z 3 +(z+r) 3 + -+|>+(n-l)r] 3 = y 3 . 



V. Bouniakowsky 252 noted the particular solution r = 2, x = n+2, 

 v = n, of (7), and that this solution leads to the second solution 



r = r = 2, x = x J ru, v = v Q +pu, 



where p and u are given by (4), and thus derived (5), etc. Starting from 

 the latter, we obtain new solutions. For w = 3, nsv/S is the cube of v : v 2 if 



The general solution of the second equation is known to be 

 z+r=(p 3 -6pg 2 ), r=(3p 2 g-2g 3 ), v z = p 

 Taking the upper signs, we see by the first condition that 



From the evident solution p' = q = w = l, we get p = Wi = 3, g=l, etc. In (2), 

 he set r = \x and noted that the rational cubic for X has no rational root 

 when n<& except for n = 3, and stated that (li) is the only solution in 

 positive cubes. 



249 Annali di Mat., (1), 5, 1862, 328. 



250 Ibid., 329. 



251 Nouv. Ann. Math., (2), 3, 1864, 176; Zeitschr. Math. Phys., 9, 1864, 284. 



252 Bull. Acad. Sc. St. Petersbourg, 8, 1865, 163-170. 



