Repunit primes: Difference between revisions
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Base 15: 3 43 73 487 2579 |
Base 15: 3 43 73 487 2579 |
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Base 16: 2</pre> |
Base 16: 2</pre> |
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''And for my own amusement, also tested up to 2700.'' |
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<pre>Base 17: 3 5 7 11 47 71 419 |
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Base 18: 2 |
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Base 19: 19 31 47 59 61 107 337 1061 |
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Base 20: 3 11 17 1487 |
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Base 21: 3 11 17 43 271 |
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Base 22: 2 5 79 101 359 857 |
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Base 23: 5 |
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Base 24: 3 5 19 53 71 653 661 |
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Base 25: |
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Base 26: 7 43 347 |
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Base 27: 3 |
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Base 28: 2 5 17 457 1423 |
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Base 29: 5 151 |
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Base 30: 2 5 11 163 569 1789 |
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Base 31: 7 17 31 |
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Base 32: |
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Base 33: 3 197 |
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Base 34: 13 1493 |
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Base 35: 313 1297 |
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Base 36: 2</pre> |
Revision as of 15:26, 23 January 2022
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 Merseene primes.)
In base three: 111, 1111111, 1111111111111, etc.
These 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 a "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)
Raku
<lang perl6>my $limit = 2700;
say "Repunit prime digits (up to $limit) in:";
.put for (2..16).hyper(:1batch).map: -> $base {
$base.fmt("Base %2d: ") ~ (1..$limit).grep(&is-prime).grep( (1 x *).parse-base($base).is-prime )
}</lang>
- Output:
Repunit prime digits (up to 2700) in: Base 2: 2 3 5 7 13 17 19 31 61 89 107 127 521 607 1279 2203 2281 Base 3: 3 7 13 71 103 541 1091 1367 1627 Base 4: 2 Base 5: 3 7 11 13 47 127 149 181 619 929 Base 6: 2 3 7 29 71 127 271 509 1049 Base 7: 5 13 131 149 1699 Base 8: 3 Base 9: Base 10: 2 19 23 317 1031 Base 11: 17 19 73 139 907 1907 2029 Base 12: 2 3 5 19 97 109 317 353 701 Base 13: 5 7 137 283 883 991 1021 1193 Base 14: 3 7 19 31 41 2687 Base 15: 3 43 73 487 2579 Base 16: 2
And for my own amusement, also tested up to 2700.
Base 17: 3 5 7 11 47 71 419 Base 18: 2 Base 19: 19 31 47 59 61 107 337 1061 Base 20: 3 11 17 1487 Base 21: 3 11 17 43 271 Base 22: 2 5 79 101 359 857 Base 23: 5 Base 24: 3 5 19 53 71 653 661 Base 25: Base 26: 7 43 347 Base 27: 3 Base 28: 2 5 17 457 1423 Base 29: 5 151 Base 30: 2 5 11 163 569 1789 Base 31: 7 17 31 Base 32: Base 33: 3 197 Base 34: 13 1493 Base 35: 313 1297 Base 36: 2