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.

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 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, ... .

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, ... .

 Factorization of decimal repunits 

Prime factors that are  are "new factors" that haven't been mentioned before. Basically, the prime factor divides Rn but does not divide Rk for all k < n. For more information, see Factorization of repunit numbers.

R1 =1R2 =R3 = · R4 =11 · R5 = · R6 =3 ·  · 11 ·  · 37R7 = · R8 =11 ·  · 101 · R9 =32 · 37 · R10 =11 · 41 · 271 · R11 = · R12 =3 · 7 · 11 · 13 · 37 · 101 · R13 = ·  · R14 =11 · 239 · 4649 · R15 =3 ·  · 37 · 41 · 271 · R16 =11 ·  · 73 · 101 · 137 · R17 = · R18 =32 · 7 · 11 · 13 ·  · 37 ·  · 333667R19 =R20 =11 · 41 · 101 · 271 ·  · 9091 · R21 =3 · 37 ·  · 239 ·  · 4649 · R22 =112 ·  ·  ·  · 21649 · 513239R29=R24 =3 · 7 · 11 · 13 · 37 · 73 · 101 · 137 · 9901 · R25 =41 · 271 ·  ·  · R26 =11 · 53 · 79 ·  · 265371653 · R27 =33 · 37 ·  · 333667 · R28 =11 ·  · 101 · 239 ·  · 4649 · 909091 · R29 = ·  ·  ·  · R30 =3 · 7 · 11 · 13 · 31 · 37 · 41 ·  ·  · 271 ·  · 9091 · 2906161

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, ... 

 Related pages 
 Repeating decimal
 Repdigit

 Footnotes 

 Notes 

 References 

 Further reading 











 Other websites 

 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

Category:Integer sequences