A unique prime is a prime
such that
is a repeating decimal with decimal
period shared with no other prime (Yates 1980). 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 (Period) Primes and the Factorization of Cyclotomic Polynomial Minus One." Math. Japonica46, 189-195,
1997.Caldwell, C. "Unique Primes." https://t5k.org/glossary/page.php?sort=UniquePrime.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 (Yates 1980). 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.
