Unique prime
| No. of known terms | 102 |
|---|---|
| Conjectured no. of terms | Infinite |
| First terms | 3, 11, 37, 101 |
| Largest known term | (10270343-1)/9 |
| OEIS index | A040017 |
In number theory, a unique prime is a certain kind of prime number. A prime p ≠ 2, 5 is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1 / p, is equal to the period length of the reciprocal of q, 1 / q. For example, 3 is the only prime with period 1, 11 is the only prime with period 2, 37 is the only prime with period 3, 101 is the only prime with period 4, so they are unique primes. In contrast, 41 and 271 both have period 5; 7 and 13 both have period 6; 239 and 4649 both have period 7; 73 and 137 both have period 8. Therefore, none of these is a unique prime. Unique primes were first described by Samuel Yates in 1980.
The above definition is related to the decimal representation of integers. Unique primes may be defined and have been studied in any numeral base.
The representation of the reciprocal of a prime number (or, more generally, an integer) p in the numeral base b is periodic of period n if
where q is a positive integer smaller than According to the summation formula of geometric series, this may be rewritten as
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