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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Unique prime ： ウィキペディア英語版 Unique prime 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''. 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. 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 :$\backslash frac\; =\; p^c$ where Φ''n''(''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 (sorted) and (ordered by period length) in OEIS) and their periods (sequence (ordered by corresponding primes) and (sorted) in OEIS) 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.) the repunit (10270343-1)/9 is the largest known probable unique prime.〔(PRP Records: Probable Primes Top 10000 )〕 In 1996 the largest ''proven'' unique prime was (101132 + 1)/10001 or, using the notation above, (99990000)141+ 1. It has 1129 digits. The record has been improved many times since then. the largest proven unique prime is $\backslash Phi\_(10)$, it has 20160 digits.〔(''The Top Twenty Unique''; Chris Caldwell )〕 ==Binary unique primes== The first unique primes in binary (base 2) are: :3, 5, 7, 11, 13, 17, 19, 31, 41, 43, 73, 127, 151, 241, 257, 331, 337, 683, ... (sequence (sorted) and (ordered by period length) in OEIS) The period length of them are: :2, 4, 3, 10, 12, 8, 18, 5, 20, 14, 9, 7, 15, 24, 16, 30, 21, 22, ... (sequence (ordered by corresponding primes) and (sorted) in OEIS) They include Fermat primes (the period length is a power of 2), Mersenne primes (the period length is a prime) and Wagstaff primes (the period length is twice an odd prime). Additionally, if ''n'' is a natural number which is not equal to 1 or 6, than at least one prime have period ''n'' in base 2, because of the Zsigmondy theorem. Besides, if ''n'' is congruent to 4 (mod 8) and ''n'' > 20, then at least two primes have period ''n'' in base 2, (Thus, ''n'' is not a unique period in base 2) because of the Aurifeuillean factorization, for example, 113 (=$\backslash Phi\_(2)$) and 29 (=$\backslash Phi\_(2)$) both have period 28 in base 2, 37 (=$\backslash Phi\_(2)$) and 109 (=$\backslash Phi\_(2)$) both have period 36 in base 2, and that 397 (=$\backslash Phi\_(2)$) and 2113 (=$\backslash Phi\_(2)$) both have period 44 in base 2, It can be shown that a prime ''p'' is of unique period ''n'' in base 2 if and only if there exists a natural number ''c'' such that :$\backslash frac\; =\; p^c$ where $\backslash Phi\_n(2)$ is the ''n''th cyclotomic polynomial at 2, because if and only if a prime ''p'' divides $\backslash frac$, then the period length of $\backslash frac$ in base 2 is ''n''. The only known values of ''n'' such that $\backslash Phi\_n(2)$ is composite but $\backslash frac$ is prime is 18, 20, 21, 54, 147, 342, 602, and 889 (If so, $\backslash Phi\_n(2)$ must have a small factor which is also a factor of ''n''), and might have other terms (However, it is a conjecture that there is no others). Thus, they are also unique period length in base 2, but the corresponding primes of them are not of the form $\backslash Phi\_n(2)$, and all other base 2 unique primes are of the form $\backslash Phi\_n(2)$. In fact, there are no primes which ''c'' > 1 (means it is a ''true power'' of ''p'') have been discovered, all known unique primes ''p'' have that ''c'' = 1. It is conjectured that all unique primes have that ''c'' = 1 (That is, all base 2 unique primes are not Wieferich primes), and it is very possible, because it's very possible that all $\backslash Phi\_n(2)$ are square-free except while ''n'' = 364 or ''n'' = 1755, if 1093 and 3511 are only two Wieferich primes (1093 and 3511 are only two known Wieferich primes, and neither 1093 nor 3511 is unique in base 2, that is, neither 364 nor 1755 is a unique period in base 2), and even if there are other Wieferich prime, they are rare! The largest known base 2 unique prime is 257885161-1, it is also the largest known prime. With an exception of Mersenne primes, the largest known probable base 2 unique prime is $\backslash frac$,〔(PRP records )〕 and the largest ''proved'' base 2 unique prime is $\backslash frac$. Besides, the largest known probable base 2 unique prime which is not Mersenne prime or Wagstaff prime is $\backslash frac$. Similar to base 10, though they are rare (but more than the case to base 10), it is conjectured strongly that there are infinitely many base 2 unique primes, because all Mersenne primes are unique in base 2, and Mersenne primes are conjectured to be infinitely. They divide none of overpseudoprimes to base 2, but every other odd prime number divide one overpseudoprime to base 2, because if and only if a conposite number can be written as $\backslash frac$, it is an overpseudoprime to base 2. There are 52 unique primes in base 2 below 264, they are: After the table, the next 10 base 2 unique prime have period length 170, 234, 158, 165, 147, 129, 184, 89, 208, and 312, and the bits of them are 65, 73, 78, 81, 82, 84, 88, 89, 96, and 97. The binary period of ''n''th prime are :2, 4, 3, 10, 12, 8, 18, 11, 28, 5, 36, 20, 14, 23, 52, 58, 60, 66, 35, 9, 39, 82, 11, 48, 100, 51, 106, 36, 28, 7, 130, 68, 138, 148, 15, 52, 162, 83, 172, 178, 180, 95, 96, 196, 99, 210, 37, 226, 76, 29, 119, 24, 50, 16, 131, 268, 135, 92, 70, 94, 292, 102, 155, 156, 316, 30, 21, 346, 348, 88, 179, 183, 372, 378, 191, 388, 44, ... (this sequence starts at ''n'' = 2, or the prime = 3) The least prime with binary period ''n'' are :1, 3, 7, 5, 31, 1, 127, 17, 73, 11, 23, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 47, 241, 601, 2731, 262657, 29, 233, 331, 2147483647, 65537, 599479, 43691, 71, 37, 223, 174763, 79, 61681, 13367, 5419, 431, 397, 631, 2796203, 2351, 97, 4432676798593, 251, ... The number of primes with binary period ''n'' are :0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 2, 3, 3, 3, 1, 3, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 2, 3, 2, 2, 1, 3, 3, 2, 3, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 5, 2, 3, 2, 4, 3, 4, 1, 2, 1, 2, 4, 2, 1, 1, 2, ... Product of primes with binary period ''n'' are (it is the primitive part of 2''n'' - 1) :1, 3, 7, 5, 31, 1, 127, 17, 73, 11, 2047, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 8388607, 241, 1082401, 2731, 262657, 3277, 536870911, 331, 2147483647, 65537, 599479, 43691, 8727391, 4033, 137438953471, 174763, 9588151, 61681, ... The binary period level of ''n''th prime are :1, 1, 2, 1, 1, 2, 1, 2, 1, 6, 1, 2, 3, 2, 1, 1, 1, 1, 2, 8, 2, 1, 8, 2, 1, 2, 1, 3, 4, 18, 1, 2, 1, 1, 10, 3, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 6, 1, 3, 8, 2, 10, 5, 16, 2, 1, 2, 3, 4, 3, 1, 3, 2, 2, 1, 11, 16, 1, 1, 4, 2, 2, 1, 1, 2, 1, 9, 2, 2, 1, 1, 10, 6, 6, 1, 2, 6, 1, 2, 1, 2, 2, 1, 3, 2, 1, 2, 1, 1, ... The least prime with binary period level ''n'' are :3, 7, 43, 113, 251, 31, 1163, 73, 397, 151, 331, 1753, 4421, 631, 3061, 257, 1429, 127, 6043, 3121, 29611, 1321, 18539, 601, 15451, 14327, 2971, 2857, 72269, 3391, 683, 2593, 17029, 2687, 42701, 11161, 13099, 1103, 71293, 13121, 17467, 2143, 83077, 25609, 5581, 5153, 26227, 2113, 51941, 2351, ... 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Unique prime」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.