Following Yates (1980), a prime  such that  is a repeating decimal with decimal period shared with no other prime is called a unique prime. For example, 3, 11, 37, and 101 are unique primes, since they are the only primes with periods one (), two (), three (), and four () respectively. On the other hand, 41 and 271 both have period five, so neither is a unique prime.

The unique primes are the primes  such that

where  is a cyclotomic polynomial,  is the period of the unique prime,  is the greatest common divisor, and  is a positive integer.

The first few unique primes are 3, 11, 37, 101, 9091, 9901, 333667, ... (OEIS A040017), which have periods 1, 2, 3, 4, 10, 12, 9, 14, 24, ... (OEIS A051627), respectively.

REFERENCES:

Caldwell, C. "Unique Primes." https://primes.utm.edu/glossary/page.php?sort=UniquePrime.

Caldwell, C. "Unique (Period) Primes and the Factorization of Cyclotomic Polynomial Minus One." Math. Japonica 46, 189-195, 1997.

Caldwell, C. and Dubner, H. "Unique Period Primes." J. Recr. Math. 29, 43-48, 1998.

Delahaye, J.-P. "Merveilleux nombres premiers." Pour la Science, p. 324, 2000.

Sloane, N. J. A. Sequences A040017 and A051627 in "The On-Line Encyclopedia of Integer Sequences."

Yates, S. "Unique Primes." Math. Mag. 53, 314, 1980.