We spent the last couple times discussing 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. Just like Pythagorean triples come, at their heart, from the observation that \(3^2+4^2=5^2\) -- 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. Mordell, an early 20th-century mathematician, proved that there are only finitely many integer solutions 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 page for some tables of what is known). It turns out that this, too, has incredibly deep connections to “elliptic curves”, and that is enough reason to study them.

However, it is interesting that there are some which are provable by more elementary means, and later 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.
• 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 last time. (He even shows that there aren't even any other rational number solutions to \(y^2-1=x^3\).)

This is also related to a very old question which was called Catalan's conjecture - namely, are there any other consecutive perfect powers other than 8 and 9?

This was called Catalan's conjecture because, as of 2002, it is Mihailescu's Theorem! See this nice overview of the history (going back as far as the 1200s).