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:

${\displaystyle R_{n}^{(b)}\equiv 1+b+b^{2}+\cdots +b^{n-1}={b^{n}-1 \over {b-1}}\qquad {\mbox{for }}|b|\geq 2,n\geq 1.}$

This means that the number R_n^(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

${\displaystyle R_{1}^{(b)}={b-1 \over {b-1}}=1\qquad {\text{and}}\qquad R_{2}^{(b)}={b^{2}-1 \over {b-1}}=b+1\qquad {\text{for}}\ |b|\geq 2.}$

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 M_n = 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 R_n but does not divide R_k for all k < n. (sequence A102380 in the OEIS)

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, ... (sequence A067063 in the OEIS)

Related pages

• Repeating decimal
• Repdigit

Footnotes

Further reading

• Beiler, Albert H. (2013) [1964], Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, Dover Recreational Math (2nd Revised ed.), New York: Dover Publications, ISBN 978-0-486-21096-4
• Dickson, Leonard Eugene; Cresse, G.H. (1999), History of the Theory of Numbers, Volume I: Divisibility and primality (2nd Reprinted ed.), Providence, RI: AMS Chelsea Publishing, ISBN 978-0-8218-1934-0
• Francis, Richard L. (1988), "Mathematical Haystacks: Another Look at Repunit Numbers", The College Mathematics Journal, 19 (3): 240–246, doi:10.1080/07468342.1988.11973120
• Gunjikar, K. R.; Kaprekar, D. R. (1939), "Theory of Demlo numbers" (PDF), Journal of the University of Bombay, VIII (3): 3–9
• Kaprekar, D. R. (1938a), "On Wonderful Demlo numbers", The Mathematics Student, 6: 68, archived from the original on 2009-02-10, retrieved 2022-03-08
• Kaprekar, D. R. (1938b), "Demlo numbers", J. Phys. Sci. Univ. Bombay, VII (3)
• Kaprekar, D. R. (1948), Demlo numbers, Devlali, India: Khareswada
• Ribenboim, Paulo (1996-02-02), The New Book of Prime Number Records, Computers and Medicine (3rd ed.), New York: Springer, ISBN 978-0-387-94457-9
• Yates, Samuel (1982), Repunits and repetends, FL: Delray Beach, ISBN 978-0-9608652-0-8

Other websites

• Eric W. Weisstein, Repunit at 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