NTIC Points on Curves

Chapter 15 Points on Curves

We have already seen a lot of the geometric viewpoint of number theory; think about Section 13.4, for instance.

The goal of the next several chapters is to examine what other questions can one ask of a purely geometric nature – or how far geometry can go in answering other questions.

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This chapter returns to the notion of finding specific types of points on graphs of number-theoretic equations. But instead of looking at lines as we did before, there are a variety of curves we can consider.

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For instance, our previous discussion about the sum of two squares was essentially interpreted as asking when the curve

x^{2} + y^{2} = n

has an (integer) lattice point on it or not. We have completely answered this question.

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But if we were considering

x^{2} + y^{2} = n

to be about a circle of radius

\sqrt{n},

then

x^{2} + 2 y^{2} = n

must be about an ellipse! Here is a visualization of points on a couple of these ellipses.

Integer points on ellipses — 🔗
Figure 15.0.1. Integer points $x^{2} + 2 y^{2} = n$ for $n = 3, 5$

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Notice that one of them has integer points, while the other does not. Try more below.

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var('x,y')
@interact
def _(n=3):
    plot1=implicit_plot(x^2+2*y^2-n, (x,-n,n), (y,-n,n), plot_points=100)
    grid_pts = [[i,j] for i in [-n..n] for j in [-n..n]]
    plot_grid_pts = points(grid_pts,rgbcolor=(0,0,0),pointsize=2)
    lattice_pts = [coords for coords in grid_pts if (2*coords[1]^2+coords[0]^2)==n]
    plot_lattice_pts = points(lattice_pts, rgbcolor = (0,0,1),pointsize=20)
    show(plot1 + plot_grid_pts + plot_lattice_pts, figsize=[5,5],aspect_ratio=1)
    pretty_print(html("The ellipse $x^2+2y^2=%s"%n))

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Questions like this are at the heart of modern number theory – plus, there are such nice pictures! It turns out this investigation will have surprising connections to calculus and group theory too.

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With that in view, you may want to try to find integer points on the following curves. Each exemplifies a type we will discuss in this chapter.

$x^{3} = y^{2} + 2$
$x^{2} + 2 y^{2} = 9$
$x^{2} - 2 y^{2} = 1$

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What we will do is to slowly try to make our way to finding integer solutions to some more difficult Diophantine equations, using an idea about rationals which simplifies Pythagorean triple geometry. We'll then return to the integer setup once we've gotten this background.

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Summary: Points on Curves

There is surprising depth, but also surprisingly accessible questions, when investigating integer and rational points on simple nonlinear curves.

We start with rational points on conics. Fact 15.1.2 gives a famous parametrization of the points on the unit circle, though we also see in Fact 15.1.8 that some conics don't have any rational points at all.
In Section 15.2 we explore a few more fun, though less crucial, cubic questions.
The next section begins our exploration of integer points, including facts such as Fact 15.3.5 about some curves with none or one.
Then in Section 15.4 the conic (quadratic) cases begin.
We use hyperbolas to bring in the wonderful geometric Algorithm 15.5.1 for using existing points to get us more and more of them.
Can this strategy be made algebraic? The final section does so, culminating in the most general proposition Fact 15.6.4 of this type we present.

The Exercises focus a lot on filling in proof details, as well as the excitement of exploring for actual integer points.