A Repunit number is a number that consists entirely of the digit "1" in a given base (typically base 10). The name "repunit" comes from "repeated unit," referring to the repeated digit 1.
The general form of a repunit number in base b (usually base 10) is:
R_n = \frac{b^n - 1}{b - 1}
Where n is the number of digits (the number of "1"s in the number), and b is the base (which is 10 for typical repunit numbers).
For base 10, the formula becomes:
R_n = \frac{10^n - 1}{9}
Examples of Repunit Numbers in Base 10:
- R1 = 1
- R2 = 11
- R3 = 111
- R4 = 1111
- R5 = 11111
Repunit Primes
A Repunit Prime is a repunit number that is also a prime number.
For example:
- R2 = 11
- R19 = 1111111111111111111
- R23 = 11111111111111111111111
As you can see, while it's easy to create repunit numbers by repeating the digit 1, most of them are not prime. Finding those that are primes becomes much more difficult as the numbers get larger.
List of Repunit Primes
All known repunit primes (as of October 2024) are as follows:
| Repunit Prime Index (n) | Repunit Prime | Number of Digits |
|---|---|---|
| 2 | 11 | 2 |
| 19 | 1111111111111111111 | 19 |
| 23 | 11111111111111111111111 | 23 |
| 317 | 111... (317 ones) ...111 | 317 |
| 1031 | 111... (1031 ones) ...111 | 1031 |
| 49081 | 111... (49081 ones) ...111 | 49081 |
| 86453 | 111... (86453 ones) ...111 | 86453 |
| 109297 | 111... (109297 ones) ...111 | 109297 |
| 270343 | 111... (270343 ones) ...111 | 270343 |
| 5794777 | 111... (5794777 ones) ...111 | 5794777 |
| 8177207 | 111... (8177207 ones) ...111 | 8177207 |
Largest Known Repunit Prime
As of now, one of the largest known repunit primes is R8177207​, which consists of 8177207 repeated 1s.
Conclusion
In conclusion, repunit primes are a fascinating type of prime number made up entirely of the digit "1." While repunit numbers are simple in appearance, very few of them are prime, making repunit primes rare and special. Despite their simplicity, finding large repunit primes is a complex task that requires significant computational effort.
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