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Section 3.5 Surprises in Integer Equations

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 \(3^2+4^2=5^2\text{.}\) This is an interesting coincidence of powers involving nearby numbers, in this case perfect squares. 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; a perfect square and a perfect cube are close together.
As usual, we can think of this graphically, using the integer lattice.
described in detail following the image
Solutions to \(x^3=y^2-1\)
Figure 3.5.1. Solutions to \(x^3=y^2-1\)
The general form \(x^3=y^2+k\) in the preceding interact can be known both as as a Bachet equation or Mordell equation. We will use the latter for the general form and reserve the former only for the special case \(k=2\text{,}\) where a cube and square are two apart.

Historical remark 3.5.2. Bachet de Méziriac.

We will learn more about Mordell in Section 15.3. André Weil in [E.5.8] describes “Claude Gaspard Bachet, sieur de Méziriac” as a “country gentleman ... no mathematician [who somehow] developed an interest in mathematical recreations”, but who in the end provided “a reliable text of Diophantus along with a mathematically sound translation and commentary.”
Just like triangles of Pythagorean triples, this equation is connected to incredibly deep mathematics. The Bachet/Mordell equation connects directly to objects called elliptic curves. Given their importance in cryptography and theory, there is plenty of reason to study such equations; for instance, see [E.4.19, Appendix A] for the connection between congruent numbers (and hence Pythagorean triples) and elliptic curves. Studying them will take us too far afield, unfortunately.
However, some equations of the form \(x^3=y^2+k\) are solvable by more elementary means. Here are some brief examples to whet your appetite; another such is Proposition 7.6.3. See Section 15.3 for more details on this independently interesting topic.

Historical remark 3.5.3. Bachet equation.

We already saw that for \(k=2\) we get the solution \(25+2=27\text{.}\) The history is interesting; Bachet himself, in his translation and commentary on Diophantus, talked about finding rational solutions to what is now ‘his’ equation. Fermat asked the English mathematician John Wallis (most famous for his infinite product for \(\pi\)
and for a nasty controversy with Thomas Hobbes
) whether there were other solutions, and implied there were no others. Euler proved this is the only solution, but using some hidden assumptions so his proof was incomplete; see Fact 15.3.5.)

Example 3.5.4.

When \(k=-1\text{,}\) Euler’s proof in 1738
that \(9-1=8\) is the only nontrivial solution is correct, however
. 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 \(x^3=y^2-1\text{,}\) all in the midst of a paper actually about demonstrating Exercise 3.6.17.)
This is also related to a very old question which was called Catalan’s conjecture, yet again related to these funny little coincidences about powers of nearby numbers. Try exploring the question with the Sage cell following it.

Question 3.5.5. Catalan’s Conjecture.

Eight and nine are consecutive perfect (nontrivial) powers. Are there any others?

Historical remark 3.5.6. Catalan’s conjecture – solved.

This was called Catalan’s conjecture because, as of 2002, the fact that there are no other such powers is Mihailescu’s Theorem! The history of this question goes back to the 1200s and Levi ben Gerson. This article by Ivars Peterson
and [E.4.18] have nice overviews of many important pieces of its history, and Wolfram MathWorld
has an accessible introduction to the mathematics.