I'd appreciate some help or at least a hint for the following exercise:
Construct a (as simple as possible) deductive system where all sequences of the form $1^n$ (which means 111... $n$-times) is provable if and only if n is prime. (Note: As simple as possible means that the deductive rules and axioms should follow a simple schema. For example the schema "$1^n$ is an axiom iff n is prime" isn't a considered simple)
At first I thought this can't be possible because I would have to construct an algorithm which gives me all prime numbers. But then I thought it should be possible to decide for every natural number if it is prime by test division. For deductive system I would take all sequences of the symbols $1$,$*$,$+$ and $=$ and with them I would like to somehow construct deductive rules which are the analogy to the test division and giving me the number if the test division only works for the sequence ($111...$) itself or the sequence $1$.