Section15.3Bachet and Mordell curves
Let's start by talking about \(y^3=x^2+2\) as a type of curve. Recall from earlier that Bachet de Méziriac first asserted this had one positive integer solution in 1621, very early in the development of modern number theory. What is it?
Fermat, Wallis, and Euler also studied this and gave various discussions and proofs of this fact. This equation is actually one of a more general class of equations called the Mordell equation: \[y^3=x^2+k\; , \;\; k\in\mathbb{Z}\] Louis Mordell, an American-born British mathematician, indeed proved some remarkable theorems about this class of equations.
Notice that Mordell's set of curves are not quadratic/conic but rather cubic, which makes them more mysterious (and, as it happens, more useful for cryptography). There is a theorem that they can only have finitely many integer points (in fact, there are even useful bounds for how many that depend only on the prime factorization of \(k\)). At the same time, they are apparently “simple” enough that they can still have infinitely many rational points; Gerd Faltings won a Fields Medal for proving that higher-degree curves cannot.
Subsection15.3.1Verifying points don't exist
Proving things about Mordell's equation is quite tricky, but once in a while there is something you can do. For instance, we can verify something we can see in the interact above.
Proof
This is a simple version of a far more general statement.
Theorem15.3.2
There is no solution to \[y^2=x^3+(M^3-N^2)\] if- \(M\equiv 2\text{ mod }(4)\),
- \(N\equiv 1\text{ mod }(2)\), and
- all prime divisors \(p\) of \(N\) are of the form \(4k+1\).
The proof basically follows the same outline, and we won't spend time proving it. There are lots of similar statements one can prove too. But there is a larger point, based on the very specific conditions on \(M\) and \(N\) Namely, if we want to prove anything about these guys with current methods we have access to, we have no hope of getting any general results.