I thought about this in a sauna at a hot spring:

Is 111 prime?

No, it is divisible by 3.

What about 1111?

This looks like a prime. It is not divisible by 3, 5, or 7. But no. It has an amazing factorization: $1111=11\cdot 101$.

What about 11111? Then I got overheated.

A number of the form $11...1$ is called a repunit. Let $R_n$ be the repunit with $n$ ones.

$R_4=1111$ is divisible by $R_2=11$. If $n$ is even, then $R_n$ is divisible by $R_2=11$.

More generally, if $n$ is composite, then $R_n$ is divisible by $R_k$, where $k$ is a factor of $n$. For example, $R_9=111,111,111$ is divisible by $R_3=111$.

Therefore, $R_n$ is prime only if $n$ is prime.

$R_2=11$ is prime, but neither $R_3=111$ nor $R_5=11111$ is. The next primes after $R_2$ are $R_{19}$ and $R_{23}$, followed by $R_{317}$.

A number like 111, 222, or 4444 is called a repdigit. Every repunit is a repdigit.

Every repdigit is divisible by a repunit, e.g., $777=7\cdot 111$, so it is not prime, except for repunit primes and the single-digit primes.

#math