置換可能素数のソースを表示

{{Expand English|date=2024年6月}}
'''置換可能素数'''（ちかんかのうそすう、{{lang-en|permutable prime}}）は、与えられた{{仮リンク|底 (記数法)|en|Radix|label=基数}}において、任意の桁の数字を[[置換 (数学)|置換]]しても[[素数]]となる素数のことである。この素数を最初に研究した{{仮リンク|ハンズ・エゴン・リチャート|en|Hans-Egon Richert}}はこれを"permutable primes"（置換可能素数）と呼んだ<ref name="Richert">{{cite journal | last=Richert | first=Hans-Egon | zbl=0054.02305 | title=On permutable primtall | journal=Norsk Matematiske Tiddskrift | volume=33 | year=1951 | pages=50–54 }}</ref>が、後に"absolute primes"（絶対素数）とも呼ばれた<ref>{{ cite journal | first1=T.N. | last1=Bhargava | first2=P.H. | last2=Doyle | title=On the existence of absolute primes | journal=Math. Mag. | volume=47 | year=1974 | page=233 | zbl=0293.10006 }}</ref>。また、"anagrammatic prime"（[[アナグラム]]素数）とも呼ばれる。

[[十進数|基数10]]においては、49,081桁以下の全ての置換可能素数が判明している。
:[[2]], [[3]], [[5]], [[7]], [[11]], [[13]], [[17]], [[31]], [[37]], [[71]], [[73]], [[79]], [[97]], [[113]], [[131]], [[199]], 311, 337, 373, 733, 919, 991, R<sub>19</sub> (1111111111111111111), R<sub>23</sub>, R<sub>317</sub>, R<sub>1031</sub>, ... {{OEIS|id=A003459}}

上記から、置換により同じ数字となるもののうち最小のもの以外を除くと、以下の16個となる。
:2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 199, 337, R<sub>19</sub>, R<sub>23</sub>, R<sub>317</sub>, R<sub>1031</sub>, ... {{OEIS|id=A258706}}

ここで、 R<sub>''n''</sub> = <math>\tfrac{10^n-1}{9}</math> は、''n''個の1（基数10）だけで構成される数（[[レピュニット|レピュニット数]]）である。全ての[[レピュニット素数]]は上記に定義した置換可能素数であるが、定義によっては少なくとも2つの異なる桁が必要となる<ref>Chris Caldwell, [http://primes.utm.edu/glossary/page.php?sort=PermutablePrime The Prime Glossary: permutable prime] at The [[Prime Pages]].</ref>。

全ての2桁以上の置換可能素数は1,3,7,9で構成されている。これは、2以外の偶数は素数ではなく、5以外の素数は5で割り切れないからである。1,3,7,9の4つの数字のうちの3つを含む置換可能素数が存在しないこと、1,3,7,9から選択された2つの数字の各々が2つ以上から構成される置換可能素数が存在しないことは証明されている<ref>A.W. Johnson, "Absolute primes," ''Mathematics Magazine'' '''50''' (1977), 100–103.</ref>。

3 < ''n'' < 6·10<sup>175</sup>となる''n''桁のレピュニット以外の置換可能素数は存在しない<ref name="Richert"/>。上記に挙げた以外のレピュニットでない置換可能素数は存在しないと[[予想 (数学)|予想]]されている。

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In base 2, only repunits can be permutable primes, because any 0 permuted to the ones place results in an even number. Therefore, the base 2 permutable primes are the [[Mersenne prime]]s. The generalization can safely be made that for any [[positional number system]], permutable primes with more than one digit can only have digits that are [[coprime]] with the [[radix]] of the number system. One-digit primes, meaning any prime below the radix, are always trivially permutable.

In [[duodecimal|base 12]], the smallest elements of the unique permutation sets of the permutable primes with fewer than 9,739 digits are known (using inverted two and three for ten and eleven, respectively)
:2, 3, 5, 7, Ɛ, R<sub>2</sub>, 15, 57, 5Ɛ, R<sub>3</sub>, 117, 11Ɛ, 555Ɛ, R<sub>5</sub>, R<sub>17</sub>, R<sub>81</sub>, R<sub>91</sub>, R<sub>225</sub>, R<sub>255</sub>, R<sub>4ᘔ5</sub>, ...

There is no ''n''-digit permutable prime in base 12 for 4 < ''n'' < 12<sup>144</sup> which is not a repunit. It is conjectured that there are no non-repunit permutable primes in base 12 other than those listed above.

In base 10 and base 12, every permutable prime is a repunit or a near-repdigit, that is, it is a permutation of the integer 
''P''(''b'', ''n'', ''x'', ''y'') = ''xxxx''...''xxxy''<sub>''b''</sub> (''n'' digits, in base ''b'')
where ''x'' and ''y'' are digits which is coprime to ''b''. Besides, ''x'' and ''y'' must be also coprime (since if there is a prime ''p'' divides both ''x'' and ''y'', then ''p'' also divides the number), so if ''x'' = ''y'', then ''x'' = ''y'' = 1. (This is not true in all bases, but exceptions are rare and could be finite in any given base; the only exceptions below 10<sup>9</sup> in bases up to 20 are: 139<sub>11</sub>, 36A<sub>11</sub>, 247<sub>13</sub>, 78A<sub>13</sub>, 29E<sub>19</sub> (M. Fiorentini, 2015).)

Let ''P''(''b'', ''n'', ''x'', ''y'') be a permutable prime in base ''b'' and let ''p'' be a prime such that ''n'' ≥ ''p''. If ''b'' is a [[primitive root modulo n|primitive root]] of ''p'', and ''p'' does not divide ''x'' or |''x'' - ''y''|, then ''n'' is a multiple of ''p'' - 1. (Since ''b'' is a primitive root mod ''p'' and ''p'' does not divide |''x'' − ''y''|, the ''p'' numbers ''xxxx''...''xxxy'', ''xxxx''...''xxyx'', ''xxxx''...''xyxx'', ..., ''xxxx''...''xyxx''...''xxxx'' (only the ''b''<sup>''p''−2</sup> digit is ''y'', others are all ''x''), ''xxxx''...''yxxx''...''xxxx'' (only the ''b''<sup>''p''−1</sup> digit is ''y'', others are all ''x''), ''xxxx''...''xxxx'' (the [[repdigit]] with ''n'' ''x''s) mod ''p'' are all different. That is, one is 0, another is 1, another is 2, ..., the other is ''p'' − 1. Thus, since the first ''p'' − 1 numbers are all primes, the last number (the repdigit with ''n'' ''x''s) must be divisible by ''p''. Since ''p'' does not divide ''x'', so ''p'' must divide the repunit with ''n'' 1s. Since ''b'' is a primitive root mod ''p'', the multiplicative order of ''n'' mod ''p'' is ''p'' − 1. Thus, ''n'' must be divisible by ''p'' − 1)

Thus, if ''b'' = 10, the digits coprime to 10 are {1, 3, 7, 9}. Since 10 is a primitive root mod 7, so if ''n'' ≥ 7, then either 7 divides ''x'' (in this case, ''x'' = 7, since ''x'' ∈ {1, 3, 7, 9}) or |''x'' − ''y''| (in this case, ''x'' = ''y'' = 1, since ''x'', ''y'' ∈ {1, 3, 7, 9}. That is, the prime is a repunit) or ''n'' is a multiple of 7 − 1 = 6. Similarly, since 10 is a primitive root mod 17, so if ''n'' ≥ 17, then either 17 divides ''x'' (not possible, since ''x'' ∈ {1, 3, 7, 9}) or |''x'' − ''y''| (in this case, ''x'' = ''y'' = 1, since ''x'', ''y'' ∈ {1, 3, 7, 9}. That is, the prime is a repunit) or ''n'' is a multiple of 17 − 1 = 16. Besides, 10 is also a primitive root mod 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, ..., so ''n'' ≥ 17 is very impossible (since for this primes ''p'', if ''n'' ≥ ''p'', then ''n'' is divisible by ''p'' − 1), and if 7 ≤ ''n'' < 17, then ''x'' = 7, or ''n'' is divisible by 6 (the only possible ''n'' is 12). If ''b'' = 12, the digits coprime to 12 are {1, 5, 7, 11}. Since 12 is a primitive root mod 5, so if ''n'' ≥ 5, then either 5 divides ''x'' (in this case, ''x'' = 5, since ''x'' ∈ {1, 5, 7, 11}) or |''x'' − ''y''| (in this case, either ''x'' = ''y'' = 1 (That is, the prime is a repunit) or ''x'' = 1, ''y'' = 11 or ''x'' = 11, ''y'' = 1, since ''x'', ''y'' ∈ {1, 5, 7, 11}.) or ''n'' is a multiple of 5 − 1 = 4. Similarly, since 12 is a primitive root mod 7, so if ''n'' ≥ 7, then either 7 divides ''x'' (in this case, ''x'' = 7, since ''x'' ∈ {1, 5, 7, 11}) or |''x'' − ''y''| (in this case, ''x'' = ''y'' = 1, since ''x'', ''y'' ∈ {1, 5, 7, 11}. That is, the prime is a repunit) or ''n'' is a multiple of 7 − 1 = 6. Similarly, since 12 is a primitive root mod 17, so if ''n'' ≥ 17, then either 17 divides ''x'' (not possible, since ''x'' ∈ {1, 5, 7, 11}) or |''x'' − ''y''| (in this case, ''x'' = ''y'' = 1, since ''x'', ''y'' ∈ {1, 5, 7, 11}. That is, the prime is a repunit) or ''n'' is a multiple of 17 − 1 = 16. Besides, 12 is also a primitive root mod 31, 41, 43, 53, 67, 101, 103, 113, 127, 137, 139, 149, 151, 163, 173, 197, ..., so ''n'' ≥ 17 is very impossible (since for this primes ''p'', if ''n'' ≥ ''p'', then ''n'' is divisible by ''p'' − 1), and if 7 ≤ ''n'' < 17, then ''x'' = 7 (in this case, since 5 does not divide ''x'' or ''x'' − ''y'', so ''n'' must be divisible by 4) or ''n'' is divisible by 6 (the only possible ''n'' is 12).
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== 脚注 ==
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