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 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 if and only if there exists a natural number c such that
where Φn(x) is the n-th cyclotomic polynomial. At present, more than fifty unique primes or probable primes are known. However, there are only twenty-three unique primes below 10100. The following table gives an overview of all 23 unique primes below 10100 (sequence A040017 in OEIS) 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 |
| 106 | 9,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091 |
| 93 | 900,900,900,900,900,900,900,900,900,900,990,990,990,990,990,990,990,990,990,991 |
| 134 | 909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091 |
| 294 | 142,857,157,142,857,142,856,999,999,985,714,285,714,285,857,142,857,142,855,714,285,571,428,571,428,572,857,143 |
| 196 | 999,999,999,999,990,000,000,000,000,099,999,999,999,999,000,000,000,000,009,999,999,999,999,900,000,000,000,001 |
The prime with period length 294 is similar to the reciprocal of 7 (0.142857142857142857...)
Just after the table, the twenty-fourth unique prime has 128 digits and period length 320. It can be written as (932032)2 + 1, where a subscript number n indicates n consecutive copies of the digit or group of digits before the subscript.
Though they are rare, based on the occurrence of repunit primes and probable primes, it is conjectured strongly that there are infinitely many unique primes. (Any repunit prime is unique.)
As of 2010 the repunit (10270343-1)/9 is the largest known probable unique prime.
In 1996 the largest proven unique prime was (101132 + 1)/10001 or, using the notation above, (99990000)141+ 1. Its reciprocal period is 2264. The record has been improved many times since then. As of 2010 the largest proven unique prime has 10,081 digits.
Famous quotes containing the words unique and/or prime:
“An absolute can only be given in an intuition, while all the rest has to do with analysis. We call intuition here the sympathy by which one is transported into the interior of an object in order to coincide with what there is unique and consequently inexpressible in it. Analysis, on the contrary, is the operation which reduces the object to elements already known.”
—Henri Bergson (1859–1941)
“If one had to worry about one’s actions in respect of other people’s ideas, one might as well be buried alive in an antheap or married to an ambitious violinist. Whether that man is the prime minister, modifying his opinions to catch votes, or a bourgeois in terror lest some harmless act should be misunderstood and outrage some petty convention, that man is an inferior man and I do not want to have anything to do with him any more than I want to eat canned salmon.”
—Aleister Crowley (1875–1947)