Unique prime
In
mathematics, a
unique prime is a certain kind of prime number. A prime
p ≠ 2, 5 is called unique
iff there is no other prime
q such that the
period length of its
reciprocal, 1 /
p, is equivalent to the period length of the reciprocal of
q, 1 /
q. Unique primes were first described by Samuel Yates in
1980.
It can be shown that a prime p is of unique period n iff there exists a natural number c such that
where Φ
n(
x) is the
n-th
cyclotomic polynomial; until today, 18 unique primes are known, and no others exist below 10
50. The following table gives an overview of all known unique primes (
Sloane's
A040017) and their periods (
Sloane's
A051627):
Period length | Prime |
1 | 3 |
2 | 11 |
3 | 37 |
4 | 101 |
10 | 9,091 |
12 | 9,901 |
9 | 333,667 |
14 | 909,091 |
24 | 99,990,001 |
36 | 999,999,000,001 |
48 | 9,999,999,900,000,001 |
38 | 909,090,909,090,909,091 |
19 | 1,111,111,111,111,111,111 |
23 | 11,111,111,111,111,111,111,111 |
39 | 900,900,900,900,990,990,990,991 |
62 | 909,090,909,090,909,090,909,090,909,091 |
120 | 100,009,999,999,899,989,999,000,000,010,001 |
150 | 10,000,099,999,999,989,999,899,999,000,000,000,100,001 |
External links