Repunit

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.Template:Refn

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:

$R_n^{(b)}\equiv 1+b+b^2+\cdots +b^{n-1}=\frac{b^n-1}{b-1}\text{for }|b|\ge 2,n\ge 1.$

This means that the number R_n^(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

$R_1^{(b)}=\frac{b-1}{b-1}=1\text{and}{R}_2^{(b)}=\frac{b^2-1}{b-1}=b+1\text{for}|b|\ge 2.$

The first of repunits in base-10 are with

1, 11, 111, 1111, 11111, 111111, ... Template:OEIS.

Base-2 repunits are also Mersenne numbers M_n = 2ⁿ − 1. They start with

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, ... Template:OEIS.

Factorization of decimal repunits

Prime factors that are Template:Color are "new factors" that haven't been mentioned before. Basically, the prime factor divides R_n but does not divide R_k for all k < n. Template:OEIS

R1 = 1 R2 = Template:Color R3 = Template:Color · Template:Color R4 = 11 · Template:Color R5 = Template:Color · Template:Color R6 = 3 · Template:Color · 11 · Template:Color · 37 R7 = Template:Color · Template:Color R8 = 11 · Template:Color · 101 · Template:Color R9 = 32 · 37 · Template:Color R10 = 11 · 41 · 271 · Template:Color

R11 = Template:Color · Template:Color R12 = 3 · 7 · 11 · 13 · 37 · 101 · Template:Color R13 = Template:Color · Template:Color · Template:Color R14 = 11 · 239 · 4649 · Template:Color R15 = 3 · Template:Color · 37 · 41 · 271 · Template:Color R16 = 11 · Template:Color · 73 · 101 · 137 · Template:Color R17 = Template:Color · Template:Color R18 = 32 · 7 · 11 · 13 · Template:Color · 37 · Template:Color · 333667 R19 = Template:Color R20 = 11 · 41 · 101 · 271 · Template:Color · 9091 · Template:Color

R21 = 3 · 37 · Template:Color · 239 · Template:Color · 4649 · Template:Color R22 = 112 · Template:Color · Template:Color · Template:Color · 21649 · 513239 R23 = Template:Color R24 = 3 · 7 · 11 · 13 · 37 · 73 · 101 · 137 · 9901 · Template:Color R25 = 41 · 271 · Template:Color · Template:Color · Template:Color R26 = 11 · 53 · 79 · Template:Color · 265371653 · Template:Color R27 = 33 · 37 · Template:Color · 333667 · Template:Color R28 = 11 · Template:Color · 101 · 239 · Template:Color · 4649 · 909091 · Template:Color R29 = Template:Color · Template:Color · Template:Color · Template:Color · Template:Color R30 = 3 · 7 · 11 · 13 · 31 · 37 · 41 · Template:Color · Template:Color · 271 · Template:Color · 9091 · 2906161

The smallest prime factors of R_n 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, ... Template:OEIS

Related pages

• Repeating decimal
• Repdigit

Further reading

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Other websites

• Template:Mathworld
• The main tables of the Cunningham project.
• Repunit at The Prime Pages by Chris Caldwell.
• Repunits and their prime factors at World!Of Numbers.
• Prime generalized repunits of at least 1000 decimal digits by Andy Steward
• Repunit Primes Project Giovanni Di Maria's repunit primes page.
• Smallest odd prime p such that (b^p-1)/(b-1) and (b^p+1)/(b+1) is prime for bases 2<=b<=1024
• Factorization of repunit numbers
• Generalized repunit primes in base -50 to 50