CHAP, xxi] RELATIONS BETWEEN FIVE OR MORE CUBES. 563 



RELATIONS BETWEEN FIVE OR MORE CUBES. 



To divide a given cube k* into n (n>2) positive cubes, J. Whitley 109 

 took a, k v, vk 2 fa- a, dv, ev, as the roots of the required cubes. Then 



3ka 3 (k 3 -a 3 ) 

 ~k*+a e (d 3 +e 3 -{ ------ 1)' 



S. Ryley took a, va, k a-v/k*, dv, ev, as the roots; then 



3k 3 a(k 3 -a 3 )=v{k G (l+d 3 +e 3 -\ ---- )-a 6 }. 



F. Elefanti 110 noted that 



9 3 = l+6 3 +8 3 , 13 3 = l+5 3 +7 3 -fl2 3 , 16 3 = 4 3 +G 3 +7 3 +9 3 +14 3 , 



and that 28 3 is a sum of 9 cubes, also of 11 cubes; etc. For the second 

 relation see Bouniakowsky 54 of Ch. VIII. 

 Y. Hirano 111 noted that 



= (36c 3 ) 3 +(a 3 ) 3 +(6 3 ) 3 +(a 3 6 3 +36c 3 ) 3 . 



A. Martin 112 noted that the sum of the cubes of rm, qrm, sm, p\q, 

 ", Pn-sq will equal the cube of sra+gr 2 /s 2 by choice of m/q. Also, 



! 3 +2 3 +4 3 +12 3 +24 3 = 25 3 , l 3 +2 3 +52 3 +216 3 = 

 S. Re"alis 113 noted that z 3 H ----- \-z\ = z* if 

 Zi, z 3 =3a/3(a-/3)+7 3 ; Z 2 , 2 4 = 



This is not the general solution since Sz; = 0. 



E. Catalan 114 noted that x 3 = Q(x-l)-+(x-2) 3 +2 gives 



Taking z = 7/4 or z = 14-6(a/6) 3 , we get a solution of X 3 +Y 3 +Z 3 = 



in positive integers. If we multiply each term by 27(:c 6 z 3 + l) 3 z 9 , combine 



the third and fourth terms and replace x 3 by x, we get 



D. S. Hart 115 found cubes whose sum is a cube by taking ! 3 -f +n 3 = 

 and seeking by trial to make S (s+m) 3 +s 3 a sum of cubes. 

 S. Tebay 116 noted that, if x = aai, y = 0,0,2, z = aa 2 , 2u 3 = n, 



(1) x 3 +y 3 +z 3 = 2u 3 



becomes a~ 3 = n~ 1 2a'i. First, solve a ? i-\-al = nr 3 -{-s 3 by setting 



109 Ladies' Diary, 1832, 41-2, Quest. 1536. 

 110 Quar. Jour. Math., 4, 1861, 339. 



111 Easy Solution of Math. Problems, 1863. Cf. Hayashi, Tohoku Math. Jour., 10, 1916, 18. 



112 Math. Quest. Educ. Times, 21, 1874, 104. 



113 Nouv. Corresp. Math., 4, 1878, 350-2. 



114 Ibid., 352-4, 371-3. 



116 Math. Quest. Educ. Times, 23, 1875, 82-3; Math. Magazine, 1, 1882-4, 173-6. 

 lia Math. Quest. Educ. Times, 38, 1883, 101-3. 



