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<caldwell@utm.edu> 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 respectivelyso 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 thirtytwo, 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 10^{47} primes below 10^{50}, only eighteen of these primes are unique primes (all listed in the table below). We can find the unique primes using the following theorem.
It is possible to generalize this to other bases, and the generalized unique primes in basex (any integer greater than one) are the prime factors of which do not divide x.
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Chris K. Caldwell © 19992014 (all rights reserved)
