If the output was plotted (from the minimum possible output to the maximum) and run a sufficient number of times, I would expect the chart of the output to have a very good distribution, and not have any serious hotspots.
To prove this, my solution would probably go something like: Probability of number N being output = 1 / MaxPossibleNumber
Then I would run a test a sufficient number of times (depends on what MaxPossibleNumber is) such that as the amount of numbers generated approaches infinity, the percentage of times any particular number is output approaches 1/MaxNumber.
The harder part would be making sure sequences of numbers don't repeat. In addition to looking at the total distribution, I think I would also want to log the order in which the numbers were generated and then run an algorithm that looks for sequence repeats.
I am NOT familiar with how random number generators work, so this is a naive solution off the top of my head and could be completely wrong, but w/e.