A unique prime is a prime number such that is a repeating decimal
with a decimal period shared with no other prime
number (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. A prime
number
is a unique prime of period iff
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. Paris:
Belin/Pour la Science, p. 324, 2000.Sloane, N. J. A.
Sequences A040017 and A051627
in "The On-Line Encyclopedia of Integer Sequences."Yates,
S. "Periods of Unique Primes." Math. Mag.53, 314, 1980.
such that 
is a repeating decimal
with a decimal period shared with no other prime
number (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. A prime
number 
is a unique prime of period 
iff
is a cyclotomic polynomial, 
is a positive integer,
is the greatest
common divisor, and 
is a positive integer. Known unique primes include
3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, ... (OEIS A040017).
Their corresponding periods are 1, 2, 3, 4, 10, 12, 9, 14, 24, ... (OEIS A051627),
respectively.
