How to define a large prime number (for example, $7919$)?

Question

The set of natural numbers may be defined using Peano Axioms:

Under this definition of natural numbers, one may define $1$ as $0^+$, $2$ as $(0^+)^+$, $7$ as $((((((0^+)^+)^+)^+)^+)^+)^+$, etc.

My question is, how does one define a large prime number such as $7919$? Surely one may define it by literally writing $0$ followed by $7919$ copies of $+$'s (how practical is this writing?), or simplify defining $$7919= (((0^+)^+)\cdots)^+$$ specifying that there are $7919$ plus symbols. But this would be a circular definition as we are trying to define $7919$ without using the number $7919$.

Since $7919$ is a prime, I failed to see anyway of defining it using multiplication as well.

How does one define the number $7919$ using precise and concise language?

Answer 1

If you have addition and multiplication we can write

$$7919=7\cdot 10^3+9\cdot 10^2+1\cdot 10^1+9.$$

All the digits can easily be defined by your plus notation and $10^n$ can be defined recursively as $10\cdot 10^{n-1}$.

Answer 2

There is absolutely no problem with defining it as $$7919=(((0^+)^+)\dots)^+.$$ Your objection seems to be writing, for example, that there are "$7919$ $+$ signs". But we don't need to do so, we can always in theory write out the finitely many plus signs to define the number. There's absolutely nothing circular here, and writing "$7919$ $+$ signs" is merely a matter of notation and does not lead to conceptual circularity.