How to apply diophantine approximation in the form of Pell's equation?

Question

so I'm really struggling to understand how Diophantine approximations are used to approximate irrationals. I'm working through a Number Theory text book and here is the question:

Use Pell's equation $x^2 - 5y^2 = 1$ to find some good rational approximations to $\sqrt{5}$.

So a solution that equation is, $x=9, y=4$. Another thing I know is Dirchlet's Approximation Theorem is the form $$ |a- b \alpha| \leq \frac{1}{b}$$. For a,b is integer and $\alpha$ is a real number.

or in the form: $$ |a-b\sqrt{N}| \leq \frac{1}{b}$$.

N is a positive integer.

But how do I go about finding the actual approximation? Any help or guidance would be greatly appreciated. Thank you.

Answer 1

You can use the Brahmagupta identity $$(a^2-nb^2)(c^2-nd^2)=(ac+nbd)^2-n(ad+bc)^2$$ once you have one solution to find an infinite series of solutions. We plug your solution in and get $$(a^2-5b^2)(9^2-5\cdot 4^2)=(9a+20b)^2-5(4a+9b)^2$$ which, with $a=9,b=4,$ gives $1=161^2-5\cdot 72^2$ and the next pair is $161,72$. We note that $$\frac 94=2.25\\ \frac {161}{72}\approx 2.236111\\\sqrt 5 \approx 2.2360680$$ so we are getting closer.

Answer 2

The "good approximations" of the sample number chosen $$ a=\sqrt 5$$ are given by using the continued fraction expansion of $\sqrt 5$, which is periodic, explicitly $\sqrt 5=[2;4,4,4,\dots]=[2;(4)]$. The first convergents are typed below with computer aid:

sage: K.<a> = QuadraticField(5)
sage: c = continued_fraction(a)
sage: c
[2; (4)*]
sage: convergents = c.convergents()
sage: [ convergents[k] for k in range(10) ]
[2,
 9/4,
 38/17,
 161/72,
 682/305,
 2889/1292,
 12238/5473,
 51841/23184,
 219602/98209,
 930249/416020]

Now it turns out that the above numbers of the shape $x/y$, with relatively prime integers $x,y>0$ are corresponding to some units $x+y\sqrt{5}=x+ay$, more exactly to those of norm equal (!) one, $$ \operatorname{Norm}(x+y\sqrt 5) = (x+y\sqrt 5)(x-y\sqrt 5) = x^2-5y^2 \overset{(!)}=1 $$ in the ring of algebraic integers $\Bbb Z[\frac 12(1+a)]$ of the field $K=\Bbb Q(a)=\Bbb Q(\sqrt 5)$. The (or a) fundamental unit of $K$ is $u=\frac 12(a+1)$, this is not so important in our case, but $v=u^3=a+2=\sqrt 5+2$ is the power of $u$ generating units in the order $\Bbb Z[a]=\Bbb Z[\sqrt 5]$. Let us see these units $1, v, v^2, v^3,\dots$ explicitly:

sage: u = K.units()[0]
sage: u
1/2*a - 1/2
sage: 1, u, u^2, u^3
(1, 1/2*a - 1/2, -1/2*a + 3/2, a - 2)
sage: v = u^-3
sage: v
a + 2
sage: [ v^k for k in range(10) ]
[1,
 a + 2,
 4*a + 9,
 17*a + 38,
 72*a + 161,
 305*a + 682,
 1292*a + 2889,
 5473*a + 12238,
 23184*a + 51841,
 98209*a + 219602]

This is then the connection with the Pell equation.

The "best approximation" (with denominators up to a given value) is insured by the theory of continued fractions.