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.
Best Answer
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.