A circular prime is a prime number that stays prime after every rotation of its digits. The number 197 is a good example: 197, 971, and 719 are all prime, so 197 is circular prime.
Definition
A circular prime is a prime number whose digit rotations are also prime numbers. A rotation moves digits around in a cycle without changing the set of digits.
How Digit Rotations Work
Digit rotation treats the decimal digits as a loop. For a three-digit number such as 197, the rotations are 197, 971, and 719. For a two-digit number such as 13, the rotations are 13 and 31.
Rotation Example
197 → 971 → 719
Since all three numbers are prime, 197 belongs to the set of circular primes.
A number does not become a circular prime just because one rotation is prime. Every rotation must pass a primality test.
Circular Prime Examples
The single-digit circular primes are 2, 3, 5, and 7. They have only one rotation, so each one stays prime by itself.
| Number | Digit Rotations | Result |
|---|---|---|
| 13 | 13, 31 | Circular prime |
| 37 | 37, 73 | Circular prime |
| 197 | 197, 971, 719 | Circular prime |
| 23 | 23, 32 | Not circular prime; 32 is composite |
| 119 | 119, 191, 911 | Not circular prime; 119 is composite |
Why Most Numbers Cannot Be Circular Prime
For a multi-digit circular prime, every rotation must avoid ending in an even digit or 5. A rotation that ends in 0, 2, 4, 6, 8, or 5 would be divisible by 2 or 5, making it composite.
Digit Filter
Except for the single-digit primes 2 and 5, a multi-digit circular prime can only use the digits 1, 3, 7, and 9. This is only a filter. It does not prove that the number is circular prime.
Divisibility by 3 also matters. Rotating digits does not change their digit sum. If the digit sum is divisible by 3, every rotation is divisible by 3. The only circular prime affected differently is the single-digit number 3 itself.
Prime Candidate Versus Circular Prime
A number can look like a possible circular prime and still fail. The digits may avoid even endings and 5, but one rotation may still have a divisor such as 7, 11, 13, or another prime factor.
Common Mistake
Using only the digits 1, 3, 7, and 9 does not prove circular primality. For example, 119 uses allowed digits, but 119 = 7 × 17, so it is composite.
Circular Primes Below 100
There are 13 circular primes below 100:
- 2
- 3
- 5
- 7
- 11
- 13
- 17
- 31
- 37
- 71
- 73
- 79
- 97
These numbers show a useful pattern: after the single-digit cases, circular primes often appear in rotation groups. For example, 13 and 31 are both circular primes because they are rotations of each other.
How Circular Primes Are Checked
Testing a circular prime has two parts. First, the number itself must be prime. Then each digit rotation must also be prime. For small numbers, trial division up to √n is enough to test each rotation.
Check Individual Rotations
For a single rotation, you can use the prime number checker to test whether that rotated number is prime before comparing the full rotation set.
Test a Rotation for Primality
Enter one rotation at a time and check whether it is prime before comparing the full set.
Why Circular Primes Matter in Number Theory
Circular primes are not a separate kind of integer system. They are ordinary prime numbers with an extra digit-rotation condition. That condition connects primality, divisibility, decimal notation, and modular arithmetic.
Prime Number
A whole number greater than 1 with exactly two positive divisors: 1 and itself.
Composite Number
A whole number greater than 1 with more than two positive divisors.
Rotation
A cyclic rearrangement of a number’s digits, such as 197 becoming 971.
FAQ About Circular Primes
What Is a Circular Prime?
A circular prime is a prime number that remains prime under every rotation of its digits. For example, 197 is circular prime because 197, 971, and 719 are all prime.
Is 13 a Circular Prime?
Yes. The rotations of 13 are 13 and 31. Both are prime, so 13 is a circular prime.
Is 23 a Circular Prime?
No. Although 23 is prime, one of its rotations is 32, which is composite. A circular prime must keep every rotation prime.
Can a Multi-Digit Circular Prime Contain 0?
No multi-digit circular prime can contain 0. Some rotation would end in 0, making that rotation divisible by 10 and therefore composite.
Why Can Multi-Digit Circular Primes Not Contain Even Digits?
If a multi-digit number contains an even digit, one rotation will end in that digit. That rotation would be even and greater than 2, so it would be composite.
Are All Numbers Made from 1, 3, 7, and 9 Circular Prime Candidates?
They are only possible candidates. A number made from those digits can still be composite, and one of its rotations can also be composite.