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Circular Prime


A prime number p is called circular if it remains prime after any cyclic permutation of its digits. An example in base-10 is 1,193 because 1,931, 9,311, and 3,119 are all primes. The first few circular primes are 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, ... (OEIS A068652).

Base-10 circular primes not contain any digit 0, 2, 4, 5, 6, or 8, since having such a digit in the units place yields a number which is necessarily divisible by either 2 or 5 (and therefore not prime).

Every prime repunit is a circular prime.

Circular primes are rare. Including only the smallest number corresponding to each cycle gives the sequence 2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, ... (OEIS A016114; Darling 2004), together with repunits R_(23), R_(317), R_(1031), R_(49081), R_(86453), R_(109297), and R_(270343) (the last several of which are probable primes).


See also

Cyclic Permutation, Permutation, Prime Number, Repunit, Truncatable Prime

Portions of this entry contributed by Christopher Stover

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References

Caldwell, C. "Circular Prime." http://primes.utm.edu/glossary/xpage/CircularPrime.html.Darling, D. The Universal Book of Mathematics from Abracadabra to Zeno's Paradoxes. Hoboken, NJ: Wiley, 2004.De Geest, P. "Circular Primes." 2011. http://www.worldofnumbers.com/circular.htm.Sloane, N. J. A. Sequences A016114 and A068652 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Stover, Christopher and Weisstein, Eric W. "Circular Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CircularPrime.html

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