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 a factorization: $1111=11\cdot 101$.

What about 11111? Then I got overheated.

Later, I became interested in palindromes in language, which led me to study numbers as strings in math.

A number of the form $11...1$ is called a repunit, denoted $R_n$, where $n$ is the number of 1s.

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 $R_3$ nor $R_5$ is. The next primes are $R_{19}$ and $R_{23}$, followed by $R_{317}$.

A number like 111, 222, or 3333 is called a repdigit. Every repunit is a repdigit.

Repdigits are not prime (except for repunit primes and single-digit primes) because they are divisible by a repunit, e.g., $777=7\cdot 111$.

#math

[^1]: Except single-digit primes.

[^2]: Except 11.