Repunit is a portmanteau of the words "repetition" and "unit", with unit being "unit value"... or in laymans terms, 1. So 1, 11, 111, 1111 & 11111 are all repunits.

Every standard integer base has repunits since every base has the digit 1. This task involves finding the repunits in different bases that are prime.

In base two, the repunits 11, 111, 11111, 1111111, etc. are prime. (These correspond to the Mersenne primes.)

In base three: 111, 1111111, 1111111111111, etc.

Repunit primes, by definition, are also circular primes.

Any repunit in any base having a composite number of digits is necessarily composite. Only repunits (in any base) having a prime number of digits might be prime.



Rather than expanding the repunit out as a giant list of 1s or converting to base 10, it is common to just list the number of 1s in the repunit; effectively the digit count. The base two repunit primes listed above would be represented as: 2, 3, 5, 7, etc.

Many of these sequences exist on OEIS, though they aren't specifically listed as "repunit prime digits" sequences.

Some bases have very few repunit primes. Bases 4, 8, and likely 16 have only one. Base 9 has none at all. Bases above 16 may have repunit primes as well... but this task is getting large enough already.



Task
 For bases 2 through 16, Find and show, here on this page, the repunit primes as digit counts, up to a limit of 1000.


Stretch
 Increase the limit to 2700 (or as high as you have patience for.)


See also
 Wikipedia: Repunit primes
 OEIS:A000043 - Mersenne exponents: primes p such that 2^p - 1 is prime. Then 2^p - 1 is called a Mersenne prime (base 2)
 OEIS:A028491 - Numbers k such that (3^k - 1)/2 is prime (base 3)
 OEIS:A004061 - Numbers n such that (5^n - 1)/4 is prime (base 5)
 OEIS:A004062 - Numbers n such that (6^n - 1)/5 is prime (base 6)
 OEIS:A004063 - Numbers k such that (7^k - 1)/6 is prime (base 7)
 OEIS:A004023 - Indices of prime repunits: numbers n such that 11...111 (with n 1's) = (10^n - 1)/9 is prime (base 10)
 OEIS:A005808 - Numbers k such that (11^k - 1)/10 is prime (base 11)
 OEIS:A004064 - Numbers n such that (12^n - 1)/11 is prime (base 12)
 OEIS:A016054 - Numbers n such that (13^n - 1)/12 is prime (base 13)
 OEIS:A006032 - Numbers k such that (14^k - 1)/13 is prime (base 14)
 OEIS:A006033 - Numbers n such that (15^n - 1)/14 is prime (base 15)
 Related task: Circular primes


 
// Repunit primes. Nigel Galloway: January 24th., 2022
let rUnitP(b:int)=let b=bigint b in primes32()|>Seq.takeWhile((>)1000)|>Seq.map(fun n->(n,((b**n)-1I)/(b-1I)))|>Seq.filter(fun(_,n)->Open.Numeric.Primes.MillerRabin.IsProbablePrime &n)|>Seq.map fst
[2..16]|>List.iter(fun n->printf $"Base %d{n}: "; rUnitP(n)|>Seq.iter(printf "%d "); printfn "")
 