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Section 15.2 A tempting cubic interlude

It is interesting that our investigation of rational points, initially motivated by integer points like Pythagorean triples, inevitably led back to integer points. Soon we will look at some remarkable properties that sets of integer points on certain curves have, and whether any such points even exist.
But before moving on, it is worth looking at some interesting tidbits relating to another type of equation, \(x^3+ay^3=b\text{.}\)
For the first example, consider that 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 \(x^3-117y^3=5\text{.}\) 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 showing 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 (you can try it in Exercise 15.7.6). (See this MathOverflow question 2  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 in [E.7.15]. Dudeney was one of the most famous puzzle constructors of a century ago, and this puzzle is a doozy.

Question 15.2.1.

Find the (rational) diameters of two spheres whose combined volume is that of two spheres of diameters one foot and two feet.
This is equivalent to finding rational points on the curve \(x^3+y^3=9\text{.}\) The puzzle itself gives the points \((1,2)\) and \((2,1)\text{,}\) 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.
Figure 15.2.2 gives a potential first step, using the tangent line to the curve at \((2,1)\text{.}\)
Finding a rational point on Dudeney’s curve
Figure 15.2.2. Finding a rational point on Dudeney’s curve
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
\begin{equation*} \left(\frac{415280564497}{348671682660},\frac{676702467503}{348671682660}\right) \end{equation*}
which does seem a bit excessive but sure is fun 3 .
There are endless variations on such questions. If we consider Dudeney’s problem as an example of summing two perfect squares to make a perfect cube, we have a more general question that Diophantus and al-Karaji explored for their rational rational solutions.
We are now ready to begin our discussion of more integer points on curves. As mentioned before, we’ll try to find integer points on the following types of curves:
  • \(x^3=y^2+2\) (sometimes called the Bachet equation)
  • \(x^2+2y^2=9\) (a well-known friend, the ellipse)
  • \(x^2-2y^2=1\) (a hyperbola with surprising connections to \(\sqrt{2}\))
For an even more fun puzzle that swept the internet a few years back, search for an answer to a question about ‘how do you find the positive integer solutions’ to \(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=4\text{,}\) based on a paper by Bremner and Macleod.