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

it becomes

-ptf) = 0. To make the constant term zero, set h — lr, p = d2; then, for

By annulling the coefficient of m, he obtained Again,

E. Barbette126 employed the first method of Martin119 to show that 33+43+53=63, l+63+83 = 93 = l+33+43+53+83,

33+43+53+83+103 = 123==C3+83+103, l+53+63+73+83+103=]33 = 53+73+93+103, 23+33+53+73+83+93+103 = 143

are the only sets of distinct cubes ^103 whose sum is a cube. R. Norrie84 would find n cubes whose sum is a cube by taking

(n;7l--p)3 = (rz0+A)3, according as n is even or odd.

A. G6rardin127 noted that the sum of the cubes of x — 1, x, x+ 1, 2/— 1, 2/, 2/+1 is of the form 3t(t~-2q) if t=x+2f, q = 3fx-l.

R. D. Carmichael128 noted that (1) has the special solution

Z = p3dr6<r3, T^p^Ger3, 2= -6pcr2, W = p3,

and obtained a set of solutions of #3+7/3+23+w3 = 3£3 involving five parameters. A special solution of £3+27/3+323 = Z3 is x, Z = 2n3::Fm3, y=m3, z = 2m?i2.

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

A. G6rardin130 derived a solution of x*+y*+z*~hv* from a given solution, and deduced a solution of

A+B+C~X+Y+Z, A3+£3+C3=X3+F3+Z3.

M. Weill131 derived a third solution x =#1+ X(x2— rci), • • • from two given solutions of 23 = t/3+23+£3+u3; likewise for ax3+byz+czz+dfi~Q.

E. Faucuembergue132 treated #3+2/3+z3 = 4w3 by setting x = 2a, 2/ = 4b+l, 2 = 4c-l, 2Z> — 2c+l=/, 6+c = ^. Then 2a3+3/2gr+4^3 = w3, which is satis-

128 Les sommes de p-i^mes pu'«sances distinctes dgales ^L une p-i6me puissance, Li^ge, 1910, 105-132.

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

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

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

130 Ibid., 22, 1915, 130-2 (error for fc=2); 23, 1916, 107-110. mNouv. Ann. Math., (4), 17, 1917, 46, 51-53.

laa I/interme'diaire des math., 24, 1917, 40.