Repunit

Repunit prime

No. of known terms	9
Conjectured no. of terms	Infinite
First terms	11, 1111111111111111111, 11111111111111111111111
Largest known term	(10270343-1)/9
OEIS index	A004022

In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — 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.

The base-b repunits are defined as (this b can be either positive or negative)

Thus, the number R_n^(b) consists of n copies of the digit 1 in base b representation. The first two repunits base b for n=1 and n=2 are

In particular, the decimal (base-10) repunits that are often referred to as simply repunits are defined as

Thus, the number R_n = R_n⁽¹⁰⁾ consists of n copies of the digit 1 in base 10 representation. The sequence of repunits base 10 starts with

Similarly, the repunits base 2 are defined as

Thus, the number R_n⁽²⁾ consists of n copies of the digit 1 in base 2 representation. In fact, the base-2 repunits are the well-known Mersenne numbers M_n = 2ⁿ − 1, they start with

(Prime factors colored red means "new factors", i. e. the prime factor divides R_n but not divides R_k for all k < n) (sequence in the OEIS)

Smallest prime factor of R_n are

The definition of repunits was motivated by recreational mathematicians looking for prime factors of such numbers.

It is easy to show that if n is divisible by a, then R_n^(b) is divisible by R_a^(b):

where ${\displaystyle \Phi _{d}(x)}$ is the ${\displaystyle d^{\mathrm {th} }}$ cyclotomic polynomial and d ranges over the divisors of n. For p prime,

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