A repunit is a number like 11, 111, or 1111. It only has the digit 1 in it. It is a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book Recreations in the Theory of Numbers.[note 1]
A repunit prime is a repunit that is also a prime number. Primes that are repunits in base-2 are Mersenne primes.
Definition
The base-b repunits can be written as this where b is the base and n is the number that you are checking in whether or not it is a repunit:
This means that the number Rn(b) is made of of of n copies of the digit 1 in base-b representation. The first two repunits base-b for n = 1 and n = 2 are
The first of repunits in base-10 are with
- 1, 11, 111, 1111, 11111, 111111, ... (sequence A002275 in the OEIS).
Base-2 repunits are also Mersenne numbers Mn = 2n − 1. They start with
- 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, ... (sequence A000225 in the OEIS).
Factorization of decimal repunits
Prime factors that are red are "new factors" that haven't been mentioned before. Basically, the prime factor divides Rn but does not divide Rk for all k < n. (sequence A102380 in the OEIS)[2]
R1 =
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1
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R2 =
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11
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R3 =
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3 · 37
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R4 =
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11 · 101
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R5 =
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41 · 271
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R6 =
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3 · 7 · 11 · 13 · 37
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R7 =
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239 · 4649
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R8 =
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11 · 73 · 101 · 137
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R9 =
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32 · 37 · 333667
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R10 =
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11 · 41 · 271 · 9091
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R11 =
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21649 · 513239
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R12 =
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3 · 7 · 11 · 13 · 37 · 101 · 9901
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R13 =
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53 · 79 · 265371653
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R14 =
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11 · 239 · 4649 · 909091
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R15 =
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3 · 31 · 37 · 41 · 271 · 2906161
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R16 =
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11 · 17 · 73 · 101 · 137 · 5882353
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R17 =
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2071723 · 5363222357
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R18 =
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32 · 7 · 11 · 13 · 19 · 37 · 52579 · 333667
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R19 =
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1111111111111111111
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R20 =
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11 · 41 · 101 · 271 · 3541 · 9091 · 27961
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R21 =
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3 · 37 · 43 · 239 · 1933 · 4649 · 10838689
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R22 =
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112 · 23 · 4093 · 8779 · 21649 · 513239
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R23 =
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11111111111111111111111
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R24 =
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3 · 7 · 11 · 13 · 37 · 73 · 101 · 137 · 9901 · 99990001
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R25 =
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41 · 271 · 21401 · 25601 · 182521213001
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R26 =
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11 · 53 · 79 · 859 · 265371653 · 1058313049
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R27 =
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33 · 37 · 757 · 333667 · 440334654777631
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R28 =
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11 · 29 · 101 · 239 · 281 · 4649 · 909091 · 121499449
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R29 =
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3191 · 16763 · 43037 · 62003 · 77843839397
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R30 =
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3 · 7 · 11 · 13 · 31 · 37 · 41 · 211 · 241 · 271 · 2161 · 9091 · 2906161
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|
The smallest prime factors of Rn for n > 1 are
- 11, 3, 11, 41, 3, 239, 11, 3, 11, 21649, 3, 53, 11, 3, 11, 2071723, 3, 1111111111111111111, 11, 3, 11, 11111111111111111111111, 3, 41, 11, 3, 11, 3191, 3, 2791, 11, 3, 11, 41, 3, 2028119, 11, 3, 11, 83, 3, 173, 11, 3, 11, 35121409, 3, 239, 11, ... (sequence A067063 in the OEIS)