This chapter has discussed linear and quadratic Diophantine equations. As you can see, even relatively simple questions become much harder once you have to restrict yourself to integer solutions. And doing it without any more tools becomes increasingly unwieldy.

But there is one final example of a question we can at least touch on. Recall that Pythagorean triples come, at their heart, from the observation that \(3^2+4^2=5^2\). This is an interesting coincidence with close numbers. So too, we can notice that \(3^2\) and \(2^3\) are only one apart, and \(5^2\) and \(3^3\) are only two units apart.

This is known as Bachet's equation or the Mordell equation. Louis Mordell, an early 20th-century mathematician, proved that there are only finitely many integer solutions to this sort of equation for a given \(k\). However, finding them all, or even some (!) turns out to be quite tricky, especially since many have no solution. See this link for some tables of what is known.

It turns out that this, too, has incredibly deep connections to a concept we will not investigate called elliptic curves; given their importance in cryptography and theory, that is enough reason to study them. However, it is independently interesting that there are some Mordell equations which are solvable by more elementary means, and in Section 15.3 there is the opportunity to do a few. Here are some examples to whet your appetite.

• The history of the solution \(25+2=27\) for \(k=2\) is interesting. Bachet himself, in his translation and commentary on Diophantus, talked about rational solutions. Fermat asked the English mathematician John Wallis (of infinite product fame) whether there were other solutions, and implied there were no others. Euler proved this, but using some hidden assumptions so that the proof was incomplete. (See Fact 15.3.4.)

• Euler's proof in 1738 that \(9-1=8\) was the only nontrivial solution to \(k=-1\), however, is correct. He uses the same method of infinite descent we saw in Proposition 3.4.11. (He even shows that there aren't even any other rational number solutions to \(y^2-1=x^3\), all in the midst of a paper actually about demonstrating Exercise 3.6.9.)

This is also related to a very old question which was called Catalan's conjecture, yet again related to these funny little coincidences. Namely:

This was called Catalan's conjecture because, as of 2002, it is Mihailescu's Theorem! The history of this question goes back to the 1200s; [C.3.17] has a nice overview of many important pieces of its history, and Wolfram MathWorld has an accessible introduction.