查看“︁欧拉伪素数”︁的源代码

{{TA|G1=Math}}

'''欧拉伪素数'''（{{lang-en|Euler pseudoprime}}）是[[伪素数]]的一种。对于奇[[合数]]''n''以及与其[[互素]]的自然数''a''，如果
: <math>a^{(n-1)/2} \equiv \pm 1\pmod{n}</math>
成立，则称''n''为关于''a''的欧拉伪素数。欧拉伪素数是[[费马伪素数]]的推广，所有欧拉伪素数同时也是费马伪素数。

与费马伪素数类似，欧拉伪素数的定义也是源于[[费马小定理]]。该定理表明，对于[[素数]]''p''以及整数''a''，有 ''a''<sup>''p''&minus;1</sup> =  1 (mod ''p'')。对大于2的素数''p''，''p''可以表示为2''q''&nbsp;+&nbsp;1 ，其中''q''为整数。于是''a''<sup>(2''q''+1)&nbsp;&minus;&nbsp;1</sup> = 1 (mod&nbsp;''p'') 成立。再进一步化简为''a''<sup>2''q''</sup>&nbsp;&minus;&nbsp;1 = 0 (mod ''p'')。通过因式分解，得到(''a''<sup>''q''</sup>&nbsp;&minus;&nbsp;1)(''a''<sup>''q''</sup> + 1) = 0 (mod ''p'')，即''a''<sup>(''p''&minus;1)/2</sup> = ±1 (mod&nbsp;''p'')。

上式可以用作概率[[素性检验]]，其可靠性是[[费马素性检验]]的两倍。不过无法基于欧拉伪素数对素数进行确定性检验，因为存在'''绝对欧拉伪素数'''，即其关于所有与其互素的数都是欧拉伪素数。绝对欧拉伪素数是[[卡迈克尔数]]（即绝对费马伪素数）的[[子集]]。最小的绝对欧拉伪素数为[[1729]] = 7×13×19。

== 示例 ==
{|class="wikitable"
|''n''
|关于''n''的欧拉伪素数
|-
|1
|9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93, 95, 99, 105, 111, 115, 117, 119, 121, 123, 125, 129, 133, 135, 141, 143, 145, 147, 153, 155, 159, 161, 165, 169, 171, 175, 177, 183, 185, 187, 189, 195, 201, 203, 205, 207, 209, 213, 215, 217, 219, 221, 225, 231, 235, 237, 243, 245, 247, 249, 253, 255, 259, 261, 265, 267, 273, 275, 279, 285, 287, 289, 291, 295, 297, 299, ...
|-
|2
|341, 561, 1105, 1729, 1905, 2047, 2465, 3277, 4033, 4681, 5461, 6601, 8321, 8481, ...
|-
|3
|121, 703, 1541, 1729, 1891, 2465, 2821, 3281, 4961, 7381, 8401, 8911, ...
|-
|4
|341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321, 8481, 8911, ...
|-
|5
|217, 781, 1541, 1729, 5461, 5611, 6601, 7449, 7813, ...
|-
|6
|185, 217, 301, 481, 1111, 1261, 1333, 1729, 2465, 2701, 3421, 3565, 3589, 3913, 5713, 6533, 8365, ...
|-
|7
|25, 325, 703, 817, 1825, 2101, 2353, 2465, 3277, 4525, 6697, 8321, ...
|-
|8
|9, 21, 65, 105, 133, 273, 341, 481, 511, 561, 585, 1001, 1105, 1281, 1417, 1541, 1661, 1729, 1905, 2047, 2465, 2501, 3201, 3277, 3641, 4033, 4097, 4641, 4681, 4921, 5461, 6305, 6533, 6601, 7161, 8321, 8481, 9265, 9709, ...
|-
|9
|91, 121, 671, 703, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601, 7381, 8401, 8911, ...
|-
|10
|9, 33, 91, 481, 657, 1233, 1729, 2821, 2981, 4187, 5461, 6533, 6541, 6601, 7777, 8149, 8401, ...
|-
|11
|133, 305, 481, 645, 793, 1729, 2047, 2257, 2465, 4577, 4921, 5041, 5185, 8113, ...
|-
|12
|65, 91, 133, 145, 247, 377, 385, 1649, 1729, 2041, 2233, 2465, 2821, 3553, 6305, 8911, 9073, ...
|-
|13
|21, 85, 105, 561, 1099, 1785, 2465, 5149, 5185, 7107, 8841, 8911, 9577, 9637, ...
|-
|14
|15, 65, 481, 781, 793, 841, 985, 1541, 2257, 2465, 2561, 2743, 3277, 5185, 5713, 6533, 6541, 7171, 7449, 7585, 8321, 9073, ...
|-
|15
|341, 1477, 1541, 1687, 1729, 1921, 3277, 6541, 9073, ...
|-
|16
|15, 85, 91, 341, 435, 451, 561, 645, 703, 1105, 1247, 1271, 1387, 1581, 1695, 1729, 1891, 1905, 2047, 2071, 2465, 2701, 2821, 3133, 3277, 3367, 3683, 4033, 4369, 4371, 4681, 4795, 4859, 5461, 5551, 6601, 6643, 7957, 8321, 8481, 8695, 8911, 9061, 9131, 9211, 9605, 9919, ...
|-
|17
|9, 91, 145, 781, 1111, 1305, 1729, 2149, 2821, 4033, 4187, 5365, 5833, 6697, 7171, ...
|-
|18
|25, 49, 65, 133, 325, 343, 425, 1105, 1225, 1369, 1387, 1729, 1921, 2149, 2465, 2977, 4577, 5725, 5833, 5941, 6305, 6517, 6601, 7345, ...
|-
|19
|9, 45, 49, 169, 343, 561, 889, 905, 1105, 1661, 1849, 2353, 2465, 2701, 3201, 4033, 4681, 5461, 5713, 6541, 6697, 7957, 8145, 8281, 8401, 9997, ...
|-
|20
|21, 57, 133, 671, 889, 1281, 1653, 1729, 1891, 2059, 2413, 2761, 3201, 5461, 5473, 5713, 5833, 6601, 6817, 7999, ...
|-
|21
|65, 221, 703, 793, 1045, 1105, 2465, 3781, 5185, 5473, 6541, 7363, 8965, 9061, ...
|-
|22
|21, 69, 91, 105, 161, 169, 345, 485, 1183, 1247, 1541, 1729, 2041, 2047, 2413, 2465, 2821, 3241, 3801, 5551, 7665, 9453, ...
|-
|23
|33, 169, 265, 341, 385, 481, 553, 1065, 1271, 1729, 2321, 2465, 2701, 2821, 3097, 4033, 4081, 4345, 4371, 4681, 5149, 6533, 6541, 7189, 7957, 8321, 8651, 8745, 8911, 9805, ...
|-
|24
|25, 175, 553, 805, 949, 1541, 1729, 1825, 1975, 2413, 2465, 2701, 3781, 4537, 6931, 7501, 9085, 9361, ...
|-
|25
|217, 561, 781, 1541, 1729, 1891, 2821, 4123, 5461, 5611, 5731, 6601, 7449, 7813, 8029, 8911, 9881, ...
|-
|26
|9, 25, 27, 45, 133, 217, 225, 475, 561, 589, 703, 925, 1065, 2465, 3325, 3385, 3565, 3825, 4741, 4921, 5041, 5425, 6697, 8029, 9073, ...
|-
|27
|65, 121, 133, 259, 341, 365, 481, 703, 1001, 1541, 1649, 1729, 1891, 2465, 2821, 2981, 2993, 3281, 4033, 4745, 4921, 4961, 5461, 6305, 6533, 7381, 7585, 8321, 8401, 8911, 9809, 9841, 9881, ...
|-
|28
|9, 27, 145, 261, 361, 529, 785, 1305, 1431, 2041, 2413, 2465, 3201, 3277, 4553, 4699, 5149, 7065, 8321, 8401, 9841, ...
|-
|29
|15, 21, 91, 105, 341, 469, 481, 793, 871, 1729, 1897, 2105, 2257, 2821, 4371, 4411, 5149, 5185, 5473, 5565, 6097, 7161, 8321, 8401, 8421, 8841, ...
|-
|30
|49, 133, 217, 341, 403, 469, 589, 637, 871, 901, 931, 1273, 1537, 1729, 2059, 2077, 2821, 3097, 3277, 4081, 4097, 5729, 6031, 6061, 6097, 6409, 6817, 7657, 8023, 8029, 8401, 9881, ...
|}

==关于''n''的最小欧拉伪素数 ==

{|class="wikitable"
|''n''
|最小欧拉伪素数
|''n''
|最小欧拉伪素数
|''n''
|最小欧拉伪素数
|''n''
|最小欧拉伪素数
|-
|1
|9
|33
|545
|65
|33
|97
|21
|-
|2
|341
|34
|21
|66
|65
|98
|9
|-
|3
|121
|35
|9
|67
|33
|99
|25
|-
|4
|341
|36
|35
|68
|25
|100
|9
|-
|5
|217
|37
|9
|69
|35
|101
|25
|-
|6
|185
|38
|39
|70
|69
|102
|133
|-
|7
|25
|39
|133
|71
|9
|103
|51
|-
|8
|9
|40
|39
|72
|85
|104
|15
|-
|9
|91
|41
|21
|73
|9
|105
|451
|-
|10
|9
|42
|451
|74
|15
|106
|15
|-
|11
|133
|43
|21
|75
|91
|107
|9
|-
|12
|65
|44
|9
|76
|15
|108
|91
|-
|13
|21
|45
|133
|77
|39
|109
|9
|-
|14
|15
|46
|9
|78
|77
|110
|111
|-
|15
|341
|47
|65
|79
|39
|111
|55
|-
|16
|15
|48
|49
|80
|9
|112
|65
|-
|17
|9
|49
|25
|81
|91
|113
|21
|-
|18
|25
|50
|21
|82
|9
|114
|115
|-
|19
|9
|51
|25
|83
|21
|115
|57
|-
|20
|21
|52
|51
|84
|85
|116
|9
|-
|21
|65
|53
|9
|85
|21
|117
|49
|-
|22
|21
|54
|55
|86
|65
|118
|9
|-
|23
|33
|55
|9
|87
|133
|119
|15
|-
|24
|25
|56
|33
|88
|87
|120
|77
|-
|25
|217
|57
|25
|89
|9
|121
|15
|-
|26
|9
|58
|57
|90
|91
|122
|33
|-
|27
|65
|59
|15
|91
|9
|123
|85
|-
|28
|9
|60
|341
|92
|21
|124
|25
|-
|29
|15
|61
|15
|93
|25
|125
|9
|-
|30
|49
|62
|9
|94
|57
|126
|25
|-
|31
|15
|63
|341
|95
|141
|127
|9
|-
|32
|25
|64
|9
|96
|65
|128
|49
|}

== 参见 ==
* [[欧拉－雅可比伪素数]]
* [[费马伪素数]]
* [[強偽質數]]
* [[卡迈克尔数]]

== 参考文献 ==
*M. Koblitz, "A Course in Number Theory and Cryptography", Springer-Verlag, 1987.
*H. Riesel, "Prime numbers and computer methods of factorisation", Birkhäuser, Boston, Mass., 1985.

{{質數}}
[[Category:伪素数]]