Re: 61. - THE SILVER CUBES

In article <104tnu5$25koi$1@dont-email.me>,
David Entwistle  <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote:

[Find two rational cubes whose sum is 17]

The first solution can be found in seconds with a fairly
straighforward program.  Obviously Dudeney didn't do that.

I had to look up how you might go about it.  It turns out

(1) There's a solution with small numerators and denominators, but
    one of them is negative;

(2) There's a recurrence relation that given one solution produces
    another.

Applying (2) to (1) gives a solution in positive rationals.

So an easier puzzle is to find (1).

The recurrence relation can be found here, but appears to have a typo
(-z^3 for -y^3):

https://web.archive.org/web/20130721093115/http://www.math.niu.edu/~rusin/known-math/95/numthy.cub

-- Richard