Last time we saw that an investigation of integers points led us through rational points back to integer points. So today, and probably a little of next time, we will look at integer lattice points exclusively. Remember the three types of curves we talked about at the very end of last time.
Let's start by talking about $y^3=x^2+2$ as a type of curve. Recall from early in the semester 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?
This equation is one of a more general type called the Mordell equation: $$y^3=x^2+k\; , \;\; k\in\mathbb{Z}$$ Mordell, an Americanborn 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, though we won't really get into that). It 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 higherdegree curves cannot.
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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 above.
Fact: There are no integer solutions to $x^3=y^27$.
Proof:
In homework there will be something similar. In fact, it turns out that there is even a more general statement; the one we just did is $M=2$, $N=1$.
Fact: There is no solution to $$y^2=x^3+(M^3N^2)$$ if
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 hopefully you get the point  if you want to prove anything about these guys with current methods we have access to, we have no hope of getting any general results.
Let's see what I mean here by returning to Bachet's original equation, $y^3=x^2+2$.
It turns out you can say that a product which is a cube is a product of cubes in this situation, but it requires some (geometrically motivated) proof, like with $\mathbb{Z}[i]$. In his 1765 "Vollständige Anleitung zur Algebra," sections 187188 and 191, he explicitly says that this just works  in any number system with $\mathbb{Z}[\sqrt{c}]$. He solves this one in section 193, and solves $x^2+4=y^3$ using the same technique in section 192, without realizing the problem.
But we shouldn't be too hard on Euler! He was one of the first people to even consider some essentially random new number system of this type. And, in 1738, he gives a correct and full proof of the observation we made a long time ago that $8$ and $9$ is the only time a perfect square is preceded by a perfect cube, which is Mordell's equation for $k=1$.
If you are interested in more information about how to prove cases of Mordell's equation, there are many good resources, including a nice one here.
On the other hand, finding lattice points on a quadratic curve is much more tractable. This is because we understand conic sections so well, after having worked with them for two thousand years!

Here we see our second prototype, $x^2+2y^2=9$. You can see that, in addition to the obvious solution where $y=0$, there is the (nearly as obvious, because the numbers are small, but still interesting) solution $x=1,y=2$.
In general, for our purposes an ellipse is special because there are only finitely many lattice points to check. So much for the computational problem  just get a fast computer!
However, I just want to mention where a general theory for such things might come from. After all, it gets harder to check with 'industrial strength' ellipses, and we want theorems.
Although it's being removed from the curriculum nowadays, there is something that often happens in high school mathematics or firstyear college calculus where you learn how to transform one conic section to another one of the same type with a matrix. In particular, we can get from the circle $x^2+y^2=9$ to $x^2+2y^2=9$ by multiplying the vector $(x,y)$ by the matrix $\pmatrix{1& 0\\ 0& 1/\sqrt{2}}$; that would not stretch the $x$axis, but shrinks in the $y$ axis by the appropriate amount.
However, one can also think of it as both conics coming from matrices. $$\pmatrix{x & y}\pmatrix{1& 0\\ 0& 1}\pmatrix{x\\ y}=x^2+y^2$$ while $$\pmatrix{x & y}\pmatrix{1& 0\\ 0& 2}\pmatrix{x\\ y}=x^2+2y^2\; .$$ Gauss was interested in extending Fermat's question  namely, what numbers are representable in these ways, as opposed to just a sum of squares? It turns out that many such quadratic forms represent the same sets of integers.
The Sage reference manual even uses our example to demonstrate this: $$\pmatrix{x & y}\pmatrix{1& 0\\ 0& 2}\pmatrix{x\\ y}=x^2+2y^2\text{ and }\pmatrix{x & y}\pmatrix{1& 1\\ 1& 3}\pmatrix{x\\ y}=x^2+2xy+3y^2$$ both would fulfill Fermat's result about primes modulo $8$ we mentioned earlier. So both should represent 11. Clearly $11=3^2+2\cdot 1^2$ works, but what about the other version?
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Looks like $x=2,y=1$ will do it. (That's because, as one of you pointed out, $$x^2+2xy+3y^2=(x+y)^2+2y^2$$ and this is a coordinate transformation.)
So there is some very deep theory there, which is another place where lie the beginnings of algebraic number theory, just like with the Gaussian integers. But we'll let it rest there.
Instead, we will continue looking for integer points on a given specific curve. Assuming that ellipses are doable by simply counting, what is next?
The parabola comes to mind. A general parabola would look like $ny=mx^2$; this can be thought of in your usual terms as $y=ax^2$ and $a=m/n$.
Then I can just check all $x\in\mathbb{Z}$ such that $n\mid mx^2$. Since $gcd(m,n)=1$ for this (lowest terms), we would just need in fact that $n\mid x^2$ (so if $n$ is prime, $n\mid x$ suffices)!
So if $y=mx^2$ for integer $m$, any $x$ will do. That makes sense; integer input better give integer output, which would be a lattice point!
If $2y=x^2$, we just look at it as $2x$, so that requiring $x$ even will give lattice points. And so on.
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So much for integer points on the parabola.
But before we go on, I want to point something very interesting out. Look at the following two setups  one where I create the line through two integer points on the conic, the other where I create the tangent line through one integer point:


Notice that in both cases you get another integer point! This is not a coincidence. In fact, there is the following fact:
Fact: The set of rational points on a conic section is an Abelian group.
However, often we will in fact get integer points. For our purposes, that means that we can try to create new points by either doing the slopethroughtwopoints thing ('adding' points) or the tangentslope thing ('doubling' a point). We will use this momentarily below.
I doesn't always work, of course, as we are only guaranteed rational points; see below where I try this on the ellipse.

But it turns out it works very nicely indeed for the last conic section, the hyperbola. So that's what we'll start with next time!
Homework:
