A Circular Prime is a special type of Prime Number. It is a number that remains prime even when its digits are rotated. For example, if you take the number 197, and rotate its digits (197 → 971 → 719), all these numbers (197, 971, and 719) are prime. So, 197 is a circular prime.
In other words, we can say, if every rotated version of a number is also a prime, then that number is called a circular prime!
The first few circular primes are:
2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, . . .
Note: Every repunit prime is a circular prime.
Examples of Circular Primes
Some examples of circular primes with all cyclic rotations are given as follows:
- 11 (Prime) (since both digits are identical, rotation doesn’t change it)
- 37 (Prime)
73 (Prime)
Since both rotations are prime, Hence 37 is a circular prime.
- 197 (Prime)
971 (Prime)
719 (Prime)
Since all rotations are prime, Hence 197 is a circular prime.
- 113 (Prime)
131 (Prime)
311 (Prime)
Since all rotations are prime, Hence 113 is a circular prime.
How to Identify Circular Primes?
To identify circular primes, we can use the following steps:
Step 1: First determine if the number is prime or not, a prime numbers has no positive divisors other than 1 and itself.
Step 2: Rotate the Digits
Once it is confirmed that the number is prime, create all the possible rotations.
Step 3 : Check Each Rotation for Primality
For each rotated number, check whether that number is prime or not.
Result: If all the rotations of the number are prime, then the original number is classified as a circular prime.
Example: Let’s check weather 113 is circular prime or not.
Ans -Yes.
Rotate digits of 113:
131, and 311
Check each rotation:
- 131 (1 × 131): Prime
- 311 (1 × 311): Prime
Since all rotations are prime, Hence 113 is a circular prime.