I'm curious, is there a list of known primes held somewhere?
https://en.wikipedia.org/wiki/List_of_prime_numbers#External...
But you couldn't hope to list all primes up to 1024 bits.
https://en.wikipedia.org/wiki/Prime-counting_function
https://en.wikipedia.org/wiki/Prime_number_theorem
For example, there are about 2⁸⁰ primes less than 10²⁶, that is, with 26 or fewer digits. (Where could you get 2⁸⁰ bits of storage, even if you could do the huge computation necessary to check each of these?) 1024-bit primes are about 308 digits long.
You can imagine trying to get an intuition for a few different kinds of things:
* Up to what point has the primality of every number been checked?
* What is the largest general-form number whose primality has been checked conclusively? (or, what is the largest-known general-form prime?)
* What's the largest general-form semiprime that has been factored into primes without foreknowledge of the factors?
* How big are the primes we use in cryptography?
* How big are the largest-known (special-form) primes?
These things are very different orders of magnitude. I'm not sure of the exact answers to the first two, but I expect the first is below 20 digits (the π(n) calculations apparently didn't check all the individual numbers). The answers to the last 3 are: 232 digits, 308 to 1233 digits, and 22338618 digits.
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.
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.
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).