The reciprocal of every
prime p (other than two and
five) has a period, that is the decimal expansion of
1/
p repeats in blocks of some set length (see the
period of a decimal expansion). This is called the
period of
the prime p. Samuel Yates defined a
unique prime
(or
unique period prime) to be a prime which has a
period that it shares with no other prime. For example: 3,
11, 37, and 101 are the only primes with periods one, two,
three, and four respectively--so they are unique primes.
But 41 and 271 both have period five, 7 and 13 both have
period six, 239 and 4649 both have period seven, and each
of 353, 449, 641, 1409, and 69857 have period thirty-two,
showing that these primes are not unique primes.
As we would expect from any object labeled "unique,"
unique primes are extremely rare. For example, even
though there are over 1047 primes below 1050,
only eighteen of these primes are unique primes. We can find the unique
primes using the following theorem.
-
Theorem.
- The prime p is a unique prime of period n
if and only if
is a power of p where
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is the nth cyclotomic polynomial.
It is possible to generalize this to other bases, and
the generalized unique primes in base-x (any integer greater than one) are the prime factors of which do not divide x.
Definitions and Notes