TITLE: Computational number theory
QUESTION [6 upvotes]: I am interested in learning computational number theory and doing some computer experiments. 
Which sort of number theory problems can be solved by using computers?  For example, is it possible to determine the ring of integers in number field extensions or find the discriminant of the extensions? 
To what extent one can use computers to solve number theory problems?
What is the best software for this purpose?  Maple, Matlab, Pari/gp or Mathematica?

REPLY [3 votes]: You might look at Bach and de Shallit's book, for one. 
In general, I'd recommend Python (a real computer language, with built-in long ints) for numerical experimentation.
Sage is an extension of Python, with many built-in number-theoretic bits of data and also algorithms.
It is true that Mathematica and other commercial/proprietary mathematics packages have widespread commercial use, but the non-public-ness of their actual algorithms, etc., is rather disappointing if one cares about what is actually happening. Bad enough that we are not able to personally witness the correctness of a large-integer computation, but even worse if we cannot even view the algorithms involved. Also, as computer languages, they have considerable failings, I think.