Before moving on, it is worth looking at some interesting tidbits relating to another type of equation, \(x^3+ay^3=b\).

Sometimes mathematicians like to explore hard questions for their own sake. Sometimes proofs are very challenging, indeed. Then again, sometimes a very easy proof is missed.

One example of this is the equation \(x^3-117y^3=5\). At one point a well-known number theorist specializing in Diophantine equations asserted this was known to have few solutions. A few years later, using field theory, this was proved.

Two years later, a note was published in an obscure Romanian journal that if one reduces the original equation modulo nine, a simple congruence is obtained which one can show has no solutions just by trying all possibilities by hand. (See this MathOverflow question for background.)

Another interesting story related to this is that of Henry Dudeney's “Puzzle of the Doctor of Physic”, related by Andrew Bremner of Arizona State University. Dudeney was on of the most famous puzzle constructors of a century ago, and this puzzle is a doozy.

Find the (rational) diameters of two spheres whose combined volume of two spheres of diameters one foot and two feet.

The puzzle itself gives the points \((1,2)\) and \((2,1)\), so the question is whether one can find any other such points. Bremner takes the reader through a geometric tour of trying to intersect this curve with various lines with rational slope in the hope of finding a proper solution to this problem. Here is a potential first step, using the tangent line to the curve at \((2,1)\).

It turns out that this point is not acceptable as a solution (why?). In fact, it takes several more steps of connecting points to arrive at a solution, namely \[\left(\frac{415280564497}{348671682660},\frac{676702467503}{348671682660}\right)\] which does seem a bit excessive but is sure fun.