Repunits: primes and passwords

A repunit is a number whose base 10 representation consists entirely of 1s. The number consisting of n 1s is denoted R_n.

Repunit primes

A repunit prime is, unsurprisingly, a repunit number which is prime. The most obvious example is R₂ = 11. Until recently the repunit numbers confirmed to be prime were R_n for n = 2, 19, 23, 317, 1031. Now the case for n = 49081 has been confirmed.

Here is the announcement. The date posted at the top of the page is from March this year, but I believe the announcement is new. Maybe the author edited an old page and didn't update the shown date.

Repunit passwords

Incidentally, I noticed a lot of repunits when I wrote about bad passwords a few days ago. That post explored a list of commonly used but broken passwords. This is the list of passwords that password cracking software will try first. The numbers R_n are part of the list for the following values of n:

1-45, 47-49, 51, 53-54, 57-60, 62, 67, 70, 72, 77, 82, 84, 147

So 46 is the smallest value of n such that R_n is not on the list. I would not recommend using R₄₆ as a password, but surprisingly there are a lot of worse choices.

The bad password file is sorted in terms of popularity, and you might expect repunits to appear in the file in order, i.e. shorter sequences first. That is sorta true overall. But you can see streaks in the plot below showing multiple runs where longer passwords are more common than shorter passwords.

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