Integers such that all Coprime and Less are Prime

Theorem
The following positive integers have the property that all positive integers less than and coprime to it, excluding $1$, are prime:
 * $1, 2, 3, 4, 6, 8, 12, 18, 24, 30$

There are no other positive integers with this property.

Proof
Let $S_n$ denote the set of all positive integers less than and coprime to $n$, excluding $1$.

Let $\map P n$ denote the propositional function:
 * All positive integers less than and coprime to $n$, excluding $1$, are prime.

We establish that $\map P n = \T$ for all the positive integers given:

From Schatunowsky's Theorem:


 * $30$ is the greatest positive integer $n$ such that $\map P n$ is true

We note that for all primes $p$ greater than $3$, $p - 1$ is composite, and so $\map P p = \F$.

The remaining composite numbers less than $30$ are investigated:

That exhausts the list.

Hence the result.