Are there more informative names?

Question

Consider this statement: "There is only one smallest prime number greater than 1000." This statement is obviously true. We can denote this number with the letter n. Then we can look up on the Internet what this number is. This is 1009 because the previous number is 997. There are two questions here:

1. Do we get new information when we find out that this number is 1009?
2. If we receive new information, then it turns out that there are more and less informative names of objects?

EDIT:

Clarifying the second question.In the example above, n and 1009 denote the same mathematical object, but the notation 1009 seems to me to give much more knowledge about the object than n.So the question is, does notation 1009 really provide more knowledge than n? For example, i can calculate the remainder of 1009 divided by 9 and get 1,but i think that cannot be done with notation n.

Answer 1

The notation you choose does not influence the amount of information you have, but may convey that information more usefully. The sentence

The smallest prime greater than 1000 leaves a remainder of 1 when divided by 9.

says what you want to say, with no notation. To convince your reader that it's true you might have to show that the number is 1009. That argument might require some notation.