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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 the decimal expansion 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 Φn(x) is the n-th cyclotomic polynomial; until today, 18 unique primes are known, and no others exist below 1050. The following table gives an overview of all known unique primes and their periods (sequence A051627 in OEIS):
| 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 |
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10-26-2009 08:16:03
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details