Solutions of two diophantine ProUems. 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, I 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 — x,m —p, the roots of the 

 required cubes, then we shall have x^-\-{p —x)^-{-(m—p)^=Spx^ — 

 8p^x-{-m^ -3m^p-\-Smp- ~a, or Spx'^ —3p^x = a — m^-{-3m'^p — 

 3mp^, hence, wehaveSGp^x^ -36p^x-\-9p* ={3p^ -6px)^ =12Qp- 

 l2pm^-\-9p^{p — 2my=a square (1). Assume 9p" {p - 2m)^ — 

 12pm^ + ]2ap^[3p{p - 2m) + 2cy =^9p~ {p -2m)^ -i-12pc{p-2m) 

 +4c^, (2), or by reduction, we have c^ -{-3pc[p —2m) = 3ap — 3pm^, 



(3), this equation is satisfied by assuming p — o" and m^-=c(2m — p) 



(c" \ 6an 



2m — ^jj .'. put c=mn and the last of these gives TO= q _i_ 3 ' 



12an'' 18n^n — 6an^ 



30an^-n^-9a^ 

 (2) we shall have 3p~ -6px=3p{p-2m) + 2c, r.x= Qn\3a +n^' 



, „^ 72071" +(9a^+w6-30an3)X(3a+w=') 



and we shall have p — x= c~'>T^ — ; — ^"v; ' 



^ Dn2(3a + n^)2 



these and the value ofm-p found above, are the roots of the sought 



cubes, which will be exhibited under a more general form by putting 



r 



n=- ' but as the reductions which this substitution requires are ob- 

 vious, we shall not insert them. If a = 4, then by assuming n=2, 



144 470 106 , . 



we shall have 57^' qn?\' ofm' for 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 x, p — x, for the roots of the cubes, we shall get (3p^ — 

 6pa;)2=:12ap— 3p*— sq., but this evidently requires one answer to 

 be found by trial, which cannot always be done, as is the case when 

 a is 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, 



