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In recreational number theory, a unique prime or unique period prime is a certain kind of prime number. A prime p ≠ 2, 3 is called unique if there is no other prime q such that the period length of the dozenal expansion of its reciprocal, 1 / p, is equal to the period length of the reciprocal of q, 1 / q. For example, E is the only prime with period 1, 11 is the only prime with period 2, 111 is the only prime with period 3, 11111 is the only prime with period 5, E0E1 is the only prime with period X, EE01 is the only prime with period 10, so they are unique primes. In contrast, 5 and 25 both have period 4, 7 and 17 both have period 6, 46E and 2X3E both have period 7, 75 and 175 both have period 8, 31 and 3X891 both have period 9, 1E and 754E2E41 both have period E. Therefore, none of these is a unique prime.

The largest known (probable) unique prime is the largest known repunit (probable) prime: (10311407−1)/E.

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