Circular Primes¶

We're only looking for circular primes below one million, which makes this search not too strenuous. Obviously, being prime is a precondition for being a circular prime, so we can start our search by only checking primes below one million.

We can then use Python's deque class to write a function that cycles through a prime number's digits to check if it is a circular prime.

In [1]:
from collections import deque

def is_circular_prime(p):
    d = deque(str(p))
    while True:
        d.rotate()
        rp = int("".join(d))
        if rp == p:
            break
        elif not is_prime(rp):
            return False
    
    return True


circular_primes = set(p for p in prime_range(1000000) if is_circular_prime(p))
print(len(circular_primes))

55

There are other optimizations you can make - for instance, every multi-digit circular prime can only be composed of the digits 1, 3, 7, and 9 (since having another digit would eventually result in a rotation that ends in an even number or 5, which obviously isn't prime) - but why bother (for this problem) when it's already this fast?

Another note - there aren't that many known circular primes, and all of the known ones greater than one million are repunits.

Relevant sequences¶

  • Circular primes: A068652

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