Where can i find resourses to study this algebraic number theory?

Question

Where can i find material to study depper the farey fractions (continued fractions)?

I triying to solve problems like these:

1.- Show that two consecutive convergent at least one of them satisfy:

$x-h_{n}/k_{n} < 1/2(k_{n})^2 $

2.- Show that $ x = a_{0} + \sum_{k=1}^\infty (-1^n)/k_{n}k_{n+1}$

And after of all, what are $k_{n}$ and $h_{n}$ in all this discussion of continued fractions?

I went to the library one day and i spent almost two hours onle searching for a book that contains these topics, and all the book i´ve found include a very little treatment.

Answer 1

This thread may help.
The subject is called "Diophantine analysis" (or "Diophantine approximation(s)") and I enjoyed much Edward Burger's introduction "Exploring the Number Jungle".
Other references may be found in Steuding's fine online course.

Answer 2

I'm not entirely sure whether any books will directly tell you how to solve those particular problems, but if you're looking for a good number theory book. Hardy & Wright's An Introduction to the Theory of Numbers may be of good help (both in the continued fractions topic and in number theory in general.)

Or for algebraic number theory, I would recommend Ireland and Rosen's A Classical Introduction to Modern Number Theory or Neukrich's Algebraic Number Theory.

Other than reading books, I think just looking up the topic through the internet or asking professors would be good ideas.