Repunits are numbers whose digits are all $1$. In general, finding the full prime factorization of a repunit is nontrivial.
Sequence A067063 in the OEIS gives the smallest prime factor of repunits. There are no $7$'s (just the number $7$, not as a digit in a different prime) that I can see in the sequence.
$7$ does divide many repunits, but always does when $3$ is also a factor, so $7$ never shows up as the smallest prime factor. The table of the first $508$ smallest prime factors of repunits shows that $7$ is not the smallest prime factor of any of those repunits.
My question is can $7$ ever be the smallest prime factor of a repunit?
I tried to find a repunit with $7$ as the smallest factor but searched very far and $3$ is always a factor whenever $7$ is.
I tried to prove that it can't be but I have no idea how.
from Hot Weekly Questions - Mathematics Stack Exchange
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