Legendre famously "proved" that 6 could not be the sum of the cubes of two rational numbers. Puzzlist Dudeney proved him wrong by producing two rational numbers the sum of whose cubes is 6.

Find the two rational numbers without researching Legendre or Dudeney.

I started with
(p/q)^3+(x/y)^3=6
x^3/y^3=(6q^3-p^3)/q^3
then I decided to try and find solution with y=q
this gives
x^3=6q^3-p^3
so then I used to following Qbasic code to find a solution for p,q,x
cls 0
for q=2 to 100
 for p=1 to int((6*q^3)^(1/3))
  x=6*q^3-p^3
  if int(x)=x then
   print p;q;x
  end if
 next p
next q
and this finds the solution
p=17 q=21 x=37
so the 2 rational numbers are 17/21 and 37/21

Posted by Daniel on 2009-09-12 11:30:37