As a source of primes? OP never said he wanted them to be random, and pi does contain every (finite) number. :P

That is not know to be the case, only conjectured.

I thought it was proven to be trancendental? Or were you saying that this property of trancendental numbers is only conjectured?

Pi is transcendental, which means that it is not a root of a non-zero polynomial with rational coefficients.

Being transcendental does not imply that a number's expansion in a given base must include every digit string. Consider the number 1/10^1! + 1/10^2! + 1/10^3! + 1/10^4! + ....

This number, whose decimal expansion is 0.110001000000000000000001... is transcendental (proven by Liouville in 1844). Its decimal expansion clearly does not contain every decimal number. It only contains the digits 0 and 1, and after the first two places never even contains consecutive 1s.

It is known that "almost all" real numbers do in fact contain in their base b expansion every sequence of base b numbers, each sequence occurring with frequency proportional to its length. These are called "normal" numbers. Very few interesting numbers (where "interesting" means that we have some reason to be interested in aside from their normality) are known to be normal, though.

More correctly, all (AFAIK) currently provable normal numbers have been designed to be normal.

Sorry to go off topic here, but can you give an example of a number that isn't interesting?

By "interesting" I mean a number that arises out of something else that one might be interested in.

For instance, consider pi. If you are interested in geometry, pi will turn up. If you are interesting in number theory, pi will turn up (e.g., it is connected to zeta functions). If you are interested in probability and statistics, pi will turn up. If you are interested in differential equations, pi will come to the party.

If you somehow have never encountered pi, I can convey it to you by telling you about one of those things. For instance, I could tell you that it is the period of the non-zero solutions of the differential equation y'' + y = 0.

An uninteresting number would be one that has no known connection to other things. If I have a particular uninteresting number, and I want to convey it to you, I'll have to just tell you the number.

A random number would almost certainly be uninteresting, such as this hex fraction, which came from /dev/urandom on my computer: 0.bfdab557104bf2d8952fb1ea0adfd732794a353d5b35d95cda927f4ad8f6dd11f11b2e968298. It is extremely unlikely that anyone has ever seen that number before. The only known thing interesting about it is that it was made specifically as an example of a number that is not otherwise interesting.

Generate some random numbers. They aren't interesting.

Pi is transcendental, but that has nothing to do with "containing every number". A transcendental number is defined as a number which is not the root of any non-zero polynomial with rational coefficients. The two properties are not related. For example, the first example of a transcendental number (Louisville numbers) isn't capable of being base-10 normal (it only contains the digits 0 and 1).

The property you're referring to is related to normality. A normal number in a base b is a number where the frequency of digits in that base approaches 1/b, but is not a rational number (and thus does not have cycles). Pi has not been proven to be normal, but if it were then it would have the property of which you speak (which is an informal property provided by normality).