Solutions of two diophantine Problems. 157 
Remarks.—This question was proposed in the Mathematical Di- 
ary, in 1832, and the above is the solution which I sent to the editor 
at that time, to be inserted in the following number of that work, but 
as the Diary has not been published since, and probably will not be 
resumed, and as several persons have expressed a desire to see the 
solution of the question, | have concluded to publish it in the Am. 
Journal of Science. 
Qu. 2. To divide any rational number into three rational cubes. 
Let a denote the given number, and x, p —«, m — p, the roots of the 
required cubes, then we shall have «*-+-(p —«x)*+(m—p)*=3pa? - 
Bp? 2+m* -3m?p+3mp? =a, or 3px? —3p?x=a--m?+3m?p— 
3mp? , hence, we have 36p? x? —36p*x+ 9p‘ =(3p* —Gpx)? =12ap— 
12pm? +9p?(p—2m)? =a square (J). Assume 9p*(p— 2m)? — 
12pm? + 12ap=[3p(p — 2m) +2c]? =9p? (p — 2m)? + 12pe( p—2m) 
+4c*, (2), or by reduction, we have c? +3pe(p —2m)=3ap—3pm°, 
c? 
(3), this equation is satisfied by assuming p=3- @ and m® =c(2m—p) 
=e(2m—5-); .". put c=mn and thie last of these gives m= 
12an$:- — 18a°n—6an* 
(3a+n*)?° ee a Rey 
(2) we shall have 3p? —6pxr=3p(p—2m) + 2c, -° 
_Gan_ 
3atn® 
hence, p= Also, by (1) and 
30an* —n°—9a? 
wm y 6n7(Batn*) 
Tan® + (9a? +n* — 30an*) xX (Ba+n*) 
6n?(3a+n*)? 
these and the value of m — p found above, are the roots of the sought 
cubes, which will be exhibited under a more general form by putting 
and we shall have p—a= 
La . . . e . . . 
n==— > but as the reductions which this substitution requires are ob- 
vious, we shall not insert them. Ifa=4, then by assuming n=2, 
hall h 144 470 106 “a 
we shall have 355° 390’ aan” or the three roots, and by adding 
their cubes, we shall find that the sum =4, as it ought to do. Cor. 
If we wish to divide any given number as a, into two cubes, then by 
assuming v, p—., for the roots of the cubes, we shall get (8p? — 
6px)? = 12ap —3p* =sq., but this evidently requires one answer to 
be found by trial, which cannot always be done, as is the case when 
ais a cube number; but if one answer can be found, then we can 
readily find as many others as we please by the ordinary methods. 
if we wish to divide a, into any number of cubes greater than two, 
