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 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.
Caldwell, C. "Unique Primes." http://primes.utm.edu/glossary/page.php?sort=UniquePrime.Caldwell, C. "Unique (Period) Primes and the Factorization of Cyclotomic Polynomial Minus
One." Math. Japonica46, 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.
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.
such that
is a cyclotomic polynomial, 
is the period of the unique prime, 
is the greatest
common divisor, and 
is a positive integer.
