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 Rn(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 Rn = Rn(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 Rn(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 Mn = 2n − 1, they start with
(Prime factors colored red means "new factors", i. e. the prime factor divides Rn but not divides Rk for all k < n) (sequence in the OEIS)
Smallest prime factor of Rn 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 Rn(b) is divisible by Ra(b):
where is the cyclotomic polynomial and d ranges over the divisors of n. For p prime,
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