Circular prime numbers are rare. A prime number is called a circular prime, if numbers obtained, after cyclic shifts (cyclic permutation) of its digits, remain prime.
Let’s take a prime number 197.
197 is a prime.
971 is also a prime. (First digit i.e. 1 is removed and added to right side of the remaining number i.e 97. so it becomes 971)
719 is also a prime. ( Again first digit i.e 9 is removed and added to right side of the remaining number i.e. 71. so it becomes 719)
Therefore 197 is a circular prime number. Because all the three numbers obtained after cyclic shifting of digits remain prime.
Let’s take another prime number 127.
127 is a prime number. 271 is also a prime number. But 712 is divisible by 2. Therefore, 127 is a prime number but is not a circular prime number, because one of the numbers obtained after cyclic permutation of digits, i.e. 172 is not a prime.
Explanation:- Cyclic permuting or shifting is a process in which the first digit from left of a number is removed and added to right side of the remaining digits. This process of removing and adding of the first digit from left is repeated until starting digit is obtained again on the left. As all the numbers generated by cyclic shifting are also prime numbers, The original prime number is called a circular prime number.
Some of the examples of circular prime numbers are as follows:-
2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933
Properties of circular prime numbers:-
- Two digits Circular prime numbers do not contain digit 0, 2, 4, 5, 6 or 8. As numbers containing such digits at its unit’s place is divisible by either 2 or 5.
- Every prime repunit is a circular prime.
- Every permutable prime is also a circular prime. But every circular prime is not a permutable prime.
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