完全循環素数とは - 完全無欠で荒唐無稽な夢

　2,5以外の素数ｐの逆数、1/pの小数を計算するとすべて循環小数となります。

1/3=0.333333....
1/7=0.14285714285714285714285714285714....
1/11=0.09090909090909090909090909090909....
1/13=0.076923076923076923076923076923077...
1/17=0.058823529411764705882352941176471....
1/19=0.052631578947368421052631578947368....

この循環の長さは、p-1の約数になります。1/7のように7-1=6となるものを完全循環素数（レプテンド素数）と呼びます。循環する桁数がp-1となるものです。
17もそうです。0.0588235294117647の１６桁が循環していますね。
　
　実は、どの素数が完全循環となるか簡単に判定できないし、出現規則も分かっていません。
アルティンの予想（高木貞治の類体論を完成させたアルティンです）というのがありまして、自然数全体に対する出現頻度予想（=0.3739558...）はあります。

　ここでは、どんなものが完全循環素数となるかを大量にリストしてみましょう。
以下、編集可能にして３７５５個をだしておきます。
　一万個の素数中に３７５５なので、アルティンの頻度には近いです。

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66463, 66491, 66509, 66541, 66553, 66593, 66617, 66629, 66697, 66701, 66713, 66739, 66749, 66821, 66851, 66863, 66943, 66949, 67033, 67057, 67061, 67073, 67103, 67139, 67181, 67189, 67211, 67217, 67247, 67273, 67339, 67343, 67349, 67421, 67429, 67433, 67447, 67499, 67531, 67537, 67567, 67577, 67589, 67607, 67619, 67709, 67777, 67807, 67819, 67829, 67891, 67901, 67939, 67943, 67967, 67979, 67993, 68059, 68087, 68099, 68113, 68141, 68171, 68207, 68219, 68447, 68473, 68491, 68501, 68539, 68543, 68567, 68581, 68633, 68659, 68669, 68687, 68699, 68743, 68767, 68771, 68777, 68819, 68863, 68891, 68897, 68899, 68903, 68909, 68927, 68993, 69011, 69019, 69029, 69073, 69109, 69143, 69149, 69193, 69221, 69233, 69247, 69257, 69263, 69313, 69341, 69371, 69379, 69383, 69389, 69463, 69473, 69491, 69497, 69539, 69593, 69623, 69661, 69691, 69697, 69737, 69739, 69767, 69779, 69821, 69829, 69833, 69847, 69859, 69899, 69931, 69941, 70019, 70061, 70139, 70141, 70181, 70183, 70207, 70223, 70229, 70297, 70379, 70381, 70393, 70423, 70429, 70451, 70457, 70459, 70487, 70549, 70571, 70583, 70589, 70607, 70619, 70621, 70663, 70709, 70753, 70783, 70793, 70823, 70891, 70901, 70913, 70937, 70949, 70979, 71069, 71143, 71153, 71167, 71171, 71257, 71261, 71263, 71287, 71327, 71339, 71341, 71353, 71411, 71419, 71429, 71473, 71527, 71537, 71549, 71593, 71633, 71647, 71663, 71699, 71713, 71741, 71777, 71789, 71807, 71887, 71909, 71941, 71971, 71983, 71993, 72047, 72073, 72101, 72103, 72109, 72167, 72221, 72223, 72229, 72251, 72269, 72287, 72353, 72367, 72379, 72383, 72421, 72461, 72469, 72497, 72503, 72617, 72623, 72647, 72673, 72701, 72739, 72767, 72817, 72869, 72931, 72949, 72953, 72977, 73019, 73061, 73091, 73127, 73189, 73259, 73309, 73327, 73331, 73379, 73417, 73421, 73433, 73553, 73571, 73583, 73589, 73607, 73673, 73709, 73727, 73783, 73819, 73823, 73847, 73859, 73939, 73943, 74017, 74021, 74051, 74099, 74131, 74143, 74149, 74167, 74177, 74189, 74219, 74257, 74297, 74353, 74377, 74381, 74411, 74419, 74509, 74527, 74531, 74567, 74623, 74699, 74713, 74771, 74821, 74857, 74869, 74873, 74891, 74897, 74903, 75011, 75017, 75029, 75149, 75167, 75193, 75217, 75223, 75269, 75337, 75353, 75367, 75377, 75407, 75503, 75527, 75539, 75541, 75553, 75571, 75583, 75617, 75659, 75703, 75731, 75743, 75767, 75793, 75821, 75833, 75869, 75913, 75931, 75937, 75941, 75979, 75983, 75989, 76091, 76099, 76103, 76207, 76259, 76261, 76303, 76343, 76367, 76379, 76421, 76423, 76463, 76487, 76541, 76607, 76651, 76673, 76697,76771, 76777, 76781, 76819, 76829, 76873, 76913, 76943, 76949, 77047,77069, 77153, 77171, 77261, 77263, 77291, 77339, 77377, 77383, 77417,77419, 77447, 77491, 77513, 77527, 77543, 77549, 77611, 77617, 77621,77647, 77687, 77699, 77713, 77731, 77783, 77983, 78007, 78059, 78101,78137, 78139, 78167, 78179, 78193, 78229, 78233, 78259, 78301,78341,78367, 78497, 78509, 78539, 78553, 78577, 78593, 78607, 78623, 78691,78713, 78737, 78779, 78781, 78857, 78887, 78901, 78977, 78979, 78989,79063, 79087, 79103, 79139, 79153, 79181, 79193, 79229, 79259, 79273,79301, 79309, 79349, 79367, 79379, 79393, 79411, 79423, 79433, 79531,79537, 79549, 79579, 79589, 79669, 79691, 79697, 79699, 79811, 79817,79823, 79829, 79847, 79873, 79901, 79903, 79939, 79943, 79967, 79979,80051, 80141, 80149, 80153, 80167, 80177, 80251, 80273, 80309, 80341,80407, 80429, 80447, 80491, 80513, 80527, 80537, 80567, 80611, 80621,80657, 80669, 80687, 80701, 80737, 80777, 80779, 80783, 80789, 80819,80833, 80863, 80897, 80909, 80953, 80989, 81017, 81019, 81023, 81047,81097, 81181, 81233, 81299, 81343, 81349, 81353, 81371, 81457, 81463,81509, 81527, 81553, 81611, 81619, 81629, 81647, 81703, 81737, 81749,81847, 81869, 81899, 81901, 81937, 81943, 81971, 82007, 82073, 82139,82153, 82171, 82189, 82193, 82217, 82223, 82261, 82301, 82339, 82349,82421, 82457, 82463, 82469, 82487, 82549, 82567, 82571, 82619, 82633,82657, 82781, 82793, 82811, 82847, 82903, 82913, 83023, 83047, 83101,83137, 83177, 83207, 83219, 83221, 83269, 83273, 83299, 83339, 83341,83383, 83407, 83417, 83423, 83459, 83497, 83537, 83579, 83617, 83621, 83663, 83777, 83833, 83873, 83891, 83903, 83939, 83983, 84011, 84017, 84047, 84059, 84131, 84137, 84143, 84179, 84221, 84223, 84263, 84299, 84313, 84349, 84377, 84389, 84407, 84421, 84457, 84509, 84589, 84629, 84659, 84673, 84691, 84701, 84713, 84731, 84737, 84793, 84857, 84859, 84869, 85087, 85109, 85193, 85223, 85247, 85259, 85313, 85381, 85429,85451, 85469, 85487, 85513, 85549, 85571, 85577, 85607, 85619, 85621, 85661, 85703, 85781, 85793, 85817, 85819, 85829, 85847, 85903, 85909, 85931, 86011, 86017, 86069, 86131, 86137, 86183, 86257, 86291, 86297,86341, 86353, 86381, 86389, 86423, 86491, 86501, 86509, 86539, 86579,86629, 86743, 86753, 86767, 86771, 86783, 86851, 86857, 86927,86939,86993, 87011, 87103, 87149, 87179, 87221, 87223, 87251, 87257, 87299,87313, 87337, 87383, 87407, 87433, 87473, 87509, 87539, 87623, 87629,87701, 87739, 87743, 87811, 87833, 87869, 87887, 87931, 87977, 88019,88177, 88223, 88259, 88261, 88301, 88337, 88339, 88379, 88411, 88423,88469, 88499, 88513, 88589, 88607, 88651, 88657, 88789, 88793, 88811,88817, 88819, 88861, 88903, 88937, 88993, 89017, 89021, 89057, 89069,89087, 89113, 89137, 89153, 89189, 89261, 89269, 89273, 89303, 89371,89381, 89393, 89417, 89459, 89501, 89513, 89527, 89567, 89611, 89633,89657, 89659, 89669, 89753, 89767, 89779, 89783, 89819, 89821, 89833,89891, 89897, 89899, 89939, 89983, 89989, 90007, 90011, 90023, 90059,90073, 90127, 90149, 90263, 90313, 90353, 90371, 90379, 90473, 90499,90527, 90617, 90647, 90659, 90697, 90703, 90709, 90731, 90749, 90793,90821, 90823, 90833, 90863, 90887, 90971, 90977, 90989, 91019, 91033,91097, 91099, 91127, 91139, 91141, 91153, 91183, 91193, 91229, 91297,91309, 91423, 91433, 91457, 91459, 91463, 91499, 91513, 91571, 91577,91583, 91673, 91691, 91753, 91771, 91811, 91823, 91873, 91943, 91967,92033, 92051, 92143, 92153, 92177, 92179, 92189, 92219, 92221, 92251,92269, 92297, 92377, 92381, 92383, 92419, 92459, 92503, 92567, 92581,92593, 92647, 92657, 92669, 92699, 92753, 92767, 92779, 92789, 92861,92927, 92993, 93047, 93059, 93097, 93113, 93131, 93139, 93179, 93229,93251, 93257, 93263, 93287, 93377, 93383, 93407, 93419, 93463, 93487,93497, 93503, 93581, 93607, 93629, 93739, 93851, 93887, 93901, 93949,93983, 94007, 94057, 94099, 94109, 94207, 94229, 94261, 94273, 94309,94331, 94343, 94349, 94379, 94433, 94447, 94463, 94513, 94531, 94583,94651, 94709, 94727, 94771, 94777, 94781, 94793, 94811, 94819, 94823,94847, 94873, 94903, 94949, 94993, 95021, 95063, 95087, 95101, 95143,95153, 95177, 95189, 95219, 95233, 95257, 95261, 95273, 95287, 95327,95339, 95393, 95419, 95429, 95527, 95549, 95581, 95621, 95629, 95633,95651, 95701, 95713, 95731, 95737, 95783, 95789, 95819, 95857, 95869,95873, 95989, 96017, 96059, 96137, 96149, 96167, 96179, 96211, 96233,96259, 96263, 96269, 96331, 96353, 96377, 96419, 96451, 96457, 96461,96469, 96497, 96527, 96553, 96581, 96589, 96697, 96703, 96737, 96749,96821, 96851, 96857, 96953, 96989, 97073, 97103, 97127, 97171, 97177,97301, 97327, 97367, 97379, 97381, 97459, 97463, 97499, 97501, 97549,97553, 97577, 97579, 97583, 97607, 97673, 97771, 97829, 97847, 97859,97931, 97943, 97967, 98011, 98017, 98057, 98179, 98207, 98251, 98257,98297, 98299, 98327, 98377, 98389, 98407, 98411, 98429, 98459, 98473,98543, 98621, 98663, 98669, 98713, 98731, 98737, 98779, 98807, 98869,98873, 98897, 98909, 98927, 98939, 98953, 98981, 98993, 99017, 99023,99103, 99109, 99131, 99137, 99139, 99181, 99233, 99251, 99257, 99259,99371, 99377, 99487, 99497, 99527, 99577, 99581, 99607, 99611, 99623,99661, 99709, 99713, 99767, 99793, 99817, 99829, 99833, 99859, 99901,99971, 99989, 100019, 100057, 100069, 100103, 100183, 100189, 100193,100313, 100343, 100379, 100411, 100417, 100447, 100459, 100469, 100501, 100537, 100673, 100703, 100741, 100811, 100823, 100829, 100847, 100937, 100943, 101021, 101063, 101113, 101141, 101149, 101183, 101221, 101273, 101377, 101383, 101419, 101429, 101501, 101503, 101513, 101527, 101531, 101537, 101581, 101611, 101663, 101701, 101737, 101741, 101771, 101789, 101807, 101833, 101863, 101869, 101873, 101891, 101939, 102019, 102023, 102059, 102101, 102149, 102217, 102251, 102259, 102301, 102337, 102367, 102407, 102451, 102461, 102497, 102499, 102503, 102539, 102607, 102611, 102647, 102811, 102859, 102913, 102953, 102983, 103007, 103069, 103087, 103091, 103099, 103177, 103217, 103291, 103349, 103393, 103421, 103423, 103451, 103457, 103549, 103553, 103577, 103583, 103619, 103651, 103657, 103669, 103687, 103811, 103913, 103967, 103979, 103993, 104033, 104047, 104059, 104087, 104113, 104149, 104179, 104183, 104207, 104297, 104309, 104327, 104381, 104393, 104417, 104459, 104473, 104513, 104537, 104543, 104549, 104579, 104651, 104659, 104701

　途中の「17389」はシャンクス（円周率を707桁計算でも途中でミス）が見つけた循環数です。

　ではこれにいったい、この完全循環素数にどんな特徴があるのでしょうか？
それが、他の素数となにか違いがあるでしょうか？
　答えはあまり芳しいものではありません。デヴィド・ウェルズの名著『プライム・ナンバーズ』にすら、これ以上のことはでてないのです（と弁解）

　【ご参考】
　参考になりそうな計算結果を提示しておきます。
まずは、オイラー関数φ(p)の約数の分布を出しておきます。
完全循環素数のオイラー関数値はp-1です。これを因数分解して、すべての因数の出現頻度を縦軸に因子を横軸したグラフです。