11 is prime, but 111 = 3 × 37 and 11111 = 41 × 271. Repeating the digit 1 does not make primality automatic. A base-10 repunit prime is a number made entirely of 1s that is also prime.
Base-10 Repunit Formula
Rn = (10n − 1) / 9
The index n is both the number of 1s and the number of decimal digits. For example, R6 = 111111.
R1 = 1 is a repunit, but it is not prime because 1 has only one positive divisor. R2 = 11 is the first repunit prime.
Why the Index n Must Be Prime
A base-10 repunit Rn can be prime only when n itself is prime. If n is composite, write n = ab with a > 1 and b > 1. Then the repunit factors as:
Composite-Index Factorization
Rab = Ra(1 + 10a + 102a + … + 10(b−1)a)
This gives Rab a nontrivial factor, so it cannot be prime. Another way to state the same divisibility rule is:
Repunit Divisibility Rule
If m divides n, then Rm divides Rn.
For example, 2 divides 6, so R2 = 11 divides R6 = 111111. Likewise, 3 divides 6, so R3 = 111 divides R6.
Prime Index Does Not Mean Prime Repunit
A prime value of n is a necessary condition, not a guarantee. The indices 3, 5 and 7 are prime, yet R3, R5 and R7 are all composite.
Small Repunits Show the Pattern
| n | Rn | Status | Factorization or Note |
|---|---|---|---|
| 1 | 1 | Neither | 1 is not prime or composite |
| 2 | 11 | Prime | First repunit prime |
| 3 | 111 | Composite | 3 × 37 |
| 4 | 1111 | Composite | 11 × 101 |
| 5 | 11111 | Composite | 41 × 271 |
| 6 | 111111 | Composite | 3 × 7 × 11 × 13 × 37 |
| 7 | 1111111 | Composite | 239 × 4649 |
| 19 | 1111111111111111111 | Prime | Second repunit prime |
The gap from R2 to R19 shows how restrictive the condition is. Prime indices such as 3, 5, 7, 11, 13 and 17 do not produce repunit primes.
Known Base-10 Repunit Primes and Probable Primes
Checked on August 25, 2026: eight entries in the currently tracked base-10 sequence have full primality proofs. Three larger entries remain probable primes rather than proven primes.
| Repunit | Digits | Current Status | Status Detail |
|---|---|---|---|
| R2 | 2 | Proven prime | 11 |
| R19 | 19 | Proven prime | 19 ones |
| R23 | 23 | Proven prime | 23 ones |
| R317 | 317 | Proven prime | 317 ones |
| R1031 | 1,031 | Proven prime | Proved in the 1980s |
| R49081 | 49,081 | Proven prime | Primality proved in 2022 |
| R86453 | 86,453 | Proven prime | Primality proved in 2023 |
| R109297 | 109,297 | Proven prime | Primality proved in 2025 |
| R270343 | 270,343 | Probable prime | No full primality proof in the current record |
| R5794777 | 5,794,777 | Probable prime | Found as a probable prime in 2021 |
| R8177207 | 8,177,207 | Probable prime | Found as a probable prime in 2021 |
The three proof upgrades in 2022, 2023 and 2025 matter because older repunit lists may still label R49081, R86453 or R109297 as probable primes. Their status has since changed to proven prime.
Proven Prime and Probable Prime Are Different Labels
A proven prime has a mathematical primality proof that can be independently verified. A probable prime has passed strong computational primality tests, but that result is not the same as a complete proof.
This distinction becomes more visible with repunits because the numbers grow directly with the index. R109297 has 109,297 decimal digits. R8177207 has 8,177,207 digits, so writing its full decimal expansion is far less useful than the compact form (108177207 − 1) / 9.
How Repunit Structure Rejects Many Candidates Early
The composite-index rule eliminates every Rn with composite n before a large primality test is needed. This greatly reduces the candidate set, but it still leaves many prime indices whose repunits turn out to be composite.
There is also a simple decimal check when n is divisible by 3. The digit sum of Rn is exactly n because the number contains n copies of the digit 1. If n is divisible by 3, then Rn is divisible by 3. R6 is an immediate example: its digit sum is 6, so 111111 is divisible by 3.
A stronger structural identity connects pairs of repunits:
Greatest Common Divisor Identity
gcd(Rm, Rn) = Rgcd(m,n)
For m = 6 and n = 9, gcd(6, 9) = 3, so gcd(R6, R9) = R3 = 111. The repeated-digit pattern therefore carries a precise divisibility structure; it is not only a visual pattern in decimal notation.
Why Large Repunits Need Specialized Primality Tests
Trial division works well for small integers because possible divisors only need to be checked up to the square root. That approach becomes impractical for repunits with tens of thousands or millions of digits.
Large candidates are first handled with specialized probable-prime testing. Turning a probable prime into a proven prime requires a primality proof, often with algorithms designed for very large integers. Elliptic curve primality proving has been used to certify large repunit primes, including R49081, R86453 and R109297.
Small examples such as 11, 111 and 11111 fit within the site’s prime number checker. Very large repunits quickly exceed the checker’s 15-digit range and require software intended for arbitrary-precision arithmetic and large-number primality testing.
Repunit Primes Have Extra Digit-Pattern Properties
Every base-10 repunit reads the same forward and backward, so every repunit prime is also a palindromic prime. Rotating its decimal digits does not change the number either, which makes every repunit prime a circular prime as well.
These properties follow automatically from the repeated 1s. They do not establish primality: a composite repunit such as 111 still has the same palindrome and rotation symmetry.
Repunit Primes in Other Bases
The idea of a repunit depends on the number base. In base b, a generalized repunit with n digits has the form:
Generalized Repunit
R(b)n = (bn − 1) / (b − 1)
Base 10 gives the familiar decimal numbers 1, 11, 111 and so on. In base 2, the same formula becomes 2n − 1. When that number is prime, it is a Mersenne prime. The decimal repunit-prime problem is therefore one member of a broader family of prime searches based on repeated-unit representations.
Are There Infinitely Many Repunit Primes?
No proof is known that there are infinitely many base-10 repunit primes, and no proof is known that only finitely many exist. The question remains open.
The known examples are too sparse to settle the issue. The condition that n must be prime removes every composite index, while many prime indices still produce composite repunits. Large surviving candidates then create a separate computational problem: passing a probable-prime test is not the same as obtaining a proof of primality.