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



A. Genocchi 253 treated (7), i. e., to make nsv a cube. Set 



m = n 2 1, s = nV 3 , s+r m=(p+g m) 3 . 

 Then 



Set 



From the resulting rational expressions for p, q we get 



which is of the same form as the initial equation ns<r = 8v 3 . Hence one 

 solution r"j s", v' leads to a second solution r, s, v, etc. But not all solutions 

 are so obtained. More convenient formulae are obtained by setting 

 r = g+z, 2v = h+pz, where r = g, 2v = his one set of solutions. 

 L. Matthiessen 254 noted the particular solutions of (7) : 



x=-2p-l, r = 2, v 



x= p l, r = l, v= p+2. 



Also that 351120 3 is a sum of k positive cubes for /c = 3, 4, 5, 6, 7, 8. 



A. B. Evans- 55 noted that the sum of the cubes of the first n 3 integers 

 is a cube only if n = l, since (n 3 -f-l)/2 is not a cube if n>l [Euler 182 on 



D. S. Hart 256 took In 1 consecutive integers, x being the middle one. 

 The sum of their cubes is (2n - I)x 3 + (2n 3 - 3n 2 +n)z. For 2n - 1 = p 3 , the 

 sum is a cube if s = x*+l(p G l}x is a CUDe - Take x = ?+y, 8s = (2?/+p 2 ) 3 ; 

 we get y and x=(p 2 1) 2 /6. For 2n cubes, add the term (z+?i) 3 . The 

 answer is now x= {(p 2 I) 2 3}/6. 



A. Martin 257 noted that the sum of the cubes of x, x+l, - , x+n* 1 

 is a cube if x (n 4 3n 3 2n 2 +4)/6. 



Hart 258 expressed the difference of 1 3 H ----- h^ 3 and (S+m) 3 S 3 as a 

 sum of cubes by trial. 



S. Realis 259 stated that z\-\ ----- \-z 3 n =(5n+3)z* has a solution with 

 21, , z n in arithmetical progression, and solutions with 2 = 1, n^2. 



A. Martin 260 proved that 1 3 +2 3 H ----- \-n* is not a cube if n>l, since 

 7i(n+l)/2=t=p 3 . For, (2n+l) 2 = 8p 3 -J-l is of the form z 3 +l = D, which 

 holds (Euler 157 ) only if x = Q, -I, 2. He listed (p. 188) sets of 20, 25 and 

 64 consecutive cubes whose sum is a cube, besides known cases. 



"*AnnaIi di Mat., 7, 1865, 151-8; Atti Accad. Pont. Nuovi Lincei, 19, 1865-6, 43-50. 



French tranel., Jour, de Math., (2), 11, 1866, 179; Sphinx-Oedipe, 4, 1909, 73-8. Ac- 



count by M. Cantor, Zeitschr. Math. Phys., 11, 1866, 248-251. 

 261 Zeitschr. Math. Phys., 13, 1868, 348-350. 

 256 Math. Quest. Educ. Times, 14, 1871, 32-33. 

 2B6 /6id., 15, 1871, 24-6 (Math. Magazine, 1, 1884, 173-6). 



267 Ibid, p. 26. Same by J. Matteson, Collection of Dioph. Problems, 1888, Probs. 4, 5. 



268 Math. Quest. Educ. Times, 23, 1875, 82-83. 



269 Nouv. Corresp. Math., 6, 1880, 525-6, 

 260 Math. Magazine, 2, 1895, 159. 



