CHAP, xxi] RELATIONS BETWEEN FIVE OR MORE CUBES. 565 



it becomes 



hm?+3(dh-bp 2 )m+3(hd 2 -pV) = 0. 



To make the constant term zero, set h = b 2 , p = d?; then, for 6 = z 3 , 



By annulling the coefficient of ra, he obtained 

 Again, 



E. Barbette 126 employed the first method of Martin 119 to show that 

 3 3 +4 3 +5 3 = 6 3 , l+6 3 +8 3 = 9 3 = l+3 3 +4 3 +5 3 +8 3 , 



l+5 3 +6 3 +7 3 +8 3 +10 3 =13 3 = 

 2 3 +3 3 +5 3 +7 3 +8 3 +9 3 +10 3 = 14 3 



are the only sets of distinct cubes s=10 3 whose sum is a cube. 

 R. Norrie 84 would find n cubes whose sum is a cube by taking 



according as n is even or odd. 



A. Gerardin 127 noted that the sum of the cubes oix l,x,x-}-l } 2f 1, 2f, 

 2/+1 is of the form 3t(P-2q} if t = x+2f, q = 3fx-l. 



R. D. Carmichael 128 noted that (1) has the special solution 



2 = p 3 db6<r 3 , ?/ = p 3 T6cr 3 , Z= Gpo" 2 , U=p 3 , 



and obtained a set of solutions of x 3 -\-y 3 -\-z 3 -{-u 3 = 3t 3 involving five param- 

 eters. A special solution of x*+2y 5 +3z z = t 3 is x, ^ = 2n 3= Fm 3 , y = m 3 , 

 z = 2mn 2 . 



The double of a cube may be a sum of four cubes. 129 



A. Gerardin 130 derived a solution of x z +y 3 +z z = hv z from a given solution, 

 and deduced a solution of 



M. Weill 131 derived a third solution x = Xi+\(x z xi), - from two given 

 solutions of x 3 = y 3 -{-z 3 -{-t 3J t-u : ''; likewise for ax 3J rby 3 + cz 3 -{-dt 3 = Q. 



E. Fauquembergue 132 treated z 3 +?/ 3 +2 3 = 4ii 3 by setting x = 2a, ?/ = 46 + l, 

 = 4c-l, 26-2c+l=/, b+c = g. Then 2a 3 +3/ 2 j/-{-4^ = < which is satis- 



126 Les sommes de p-iemes puissances distinctes egales a une p-ieme puissance, Liege, 1910, 



105-132. 



127 L'intermediaire des math., 19, 1912, 136. 



128 Amer. Math. Monthly, 20, 1913, 304-6. 



129 L'intermediaire des math., 21, 1914, 144, 188-190; 22, 1915, 60. 



130 Ibid., 22, 1915, 130-2 (error for h = 2); 23, 1916, 107-110. 



131 Nouv. Ann. Math., (4), 17, 1917, 46, 51-53. 



132 L'intermediaire des math., 24, 1917, 40. 



