49 lines
2.7 KiB
Plaintext
49 lines
2.7 KiB
Plaintext
[[wp:Repunit|Repunit]] is a [[wp:Portmanteau|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 [[wp:Mersenne_prime|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 '''1'''s or converting to base 10, it is common to just list the ''number'' of '''1'''s 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:|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
|
|
|
|
;* [[wp:Repunit#Repunit_primes|Wikipedia: Repunit primes]]
|
|
;* [[oeis:A000043|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|OEIS:A028491 - Numbers k such that (3^k - 1)/2 is prime]] (base 3)
|
|
;* [[oeis:A004061|OEIS:A004061 - Numbers n such that (5^n - 1)/4 is prime]] (base 5)
|
|
;* [[oeis:A004062|OEIS:A004062 - Numbers n such that (6^n - 1)/5 is prime]] (base 6)
|
|
;* [[oeis:A004063|OEIS:A004063 - Numbers k such that (7^k - 1)/6 is prime]] (base 7)
|
|
;* [[oeis:A004023|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|OEIS:A005808 - Numbers k such that (11^k - 1)/10 is prime]] (base 11)
|
|
;* [[oeis:A004064|OEIS:A004064 - Numbers n such that (12^n - 1)/11 is prime]] (base 12)
|
|
;* [[oeis:A016054|OEIS:A016054 - Numbers n such that (13^n - 1)/12 is prime]] (base 13)
|
|
;* [[oeis:A006032|OEIS:A006032 - Numbers k such that (14^k - 1)/13 is prime]] (base 14)
|
|
;* [[oeis:A006033|OEIS:A006033 - Numbers n such that (15^n - 1)/14 is prime]] (base 15)
|
|
;* [[Circular primes|Related task: Circular primes]]
|
|
|
|
|
|
|