歐拉-雅可比偽素數

歐拉－雅可比偽素數（英語：Euler–Jacobi pseudoprime）是偽素數的一種。對於奇合數n以及與其互素的自然數a，如果

${\displaystyle a^{(n-1)/2}\equiv \left({\frac {a}{n}}\right){\pmod {n}}}$

成立（其中${\displaystyle \left({\frac {a}{n}}\right)}$為雅可比符號），則稱n為關於a的歐拉－雅可比偽素數，或簡稱為歐拉偽素數。

歐拉－雅可比偽素數是歐拉偽素數的推廣，所有歐拉－雅可比偽素數同時也是費馬偽素數與歐拉偽素數。由於上式對所有素數都成立，因而可以用其進行概率素性檢驗，其可靠性是費馬素性檢驗的兩倍多。此外，與絕對費馬偽素數（卡邁克爾數）與絕對歐拉偽素數不同的是，不存在絕對歐拉－雅可比偽素數，即不存在關於所有與n互素的a都是歐拉－雅可比偽素數的n。可以證明，對於n，至少存在n/2個小於n的a，n不是歐拉－雅可比偽素數。

關於2的最小歐拉－雅可比偽素數是561。而在小於25·10⁹的數中，共有11347個關於2的歐拉－雅可比偽素數（參見 A047713)）。^[1]

示例

n	關於n的歐拉－雅可比偽素數
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	561, 1105, 1729, 1905, 2047, 2465, 3277, 4033, 4681, 6601, 8321, 8481, 10585, 12801, 15841, 16705, 18705, 25761, 29341, 30121, 33153, 34945, 41041, 42799, 46657, 49141, 52633, 62745, 65281, 74665, 75361, 80581, 85489, 87249, 88357, 90751, ...
3	121, 703, 1729, 1891, 2821, 3281, 7381, 8401, 8911, 10585, 12403, 15457, 15841, 16531, 18721, 19345, 23521, 24661, 28009, 29341, 31621, 41041, 44287, 46657, 47197, 49141, 50881, 52633, 55969, 63139, 63973, 74593, 75361, 79003, 82513, 87913, 88573, 93961, 97567, ...
4	341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321, 8481, 8911, 10261, 10585, 11305, 12801, 13741, 13747, 13981, 14491, 15709, 15841, 16705, 18705, 18721, 19951, 23001, 23377, 25761, 29341, 30121, 30889, 31417, 31609, 31621, 33153, 34945, 35333, 39865, 41041, 41665, 42799, 46657, 49141, 49981, 52633, 55245, 57421, 60701, 60787, 62745, 63973, 65077, 65281, 68101, 72885, 74665, 75361, 80581, 83333, 83665, 85489, 87249, 88357, 88561, 90751, 91001, 93961, ...
5	781, 1541, 1729, 5461, 5611, 6601, 7449, 7813, 11041, 12801, 13021, 14981, 15751, 15841, 21361, 24211, 25351, 29539, 38081, 40501, 41041, 44801, 47641, 53971, 67921, 75361, 79381, 90241, ...
6	217, 481, 1111, 1261, 1729, 2701, 3589, 3913, 5713, 6533, 10585, 11041, 11137, 14701, 15841, 17329, 18361, 20017, 21049, 29341, 34441, 39493, 41041, 43621, 46657, 46873, 49141, 49321, 49661, 52633, 54481, 58969, 74023, 74563, 75361, 76921, 83333, 83665, 87061, 88561, 92053, 94657, 94697, 97751, 97921, ...
7	25, 325, 703, 2101, 2353, 2465, 3277, 4525, 11041, 13665, 14089, 19345, 20197, 29857, 29891, 38081, 39331, 46657, 49241, 58825, 64681, 76627, 78937, 79381, 87673, 88399, 88831, 89961, 92929, ...
8	9, 65, 105, 273, 481, 511, 561, 585, 1001, 1105, 1281, 1417, 1729, 1905, 2047, 2465, 2501, 3201, 3277, 3641, 4033, 4097, 4641, 4681, 4921, 6305, 6601, 7161, 8321, 8481, 9265, 10585, 10745, 11041, 12545, 12801, 13833, 14497, 15665, 15841, 16589, 16705, 16881, 17865, 18705, 19345, 19561, 20801, 23241, 24311, 24929, 25761, 29341, 30121, 32865, 33153, 33201, 34881, 34945, 35113, 37401, 38081, 40833, 41041, 41441, 42799, 43745, 45761, 46657, 49141, 49601, 50881, 52429, 52521, 52633, 52801, 54161, 55537, 55969, 56033, 57681, 59291, 59641, 61337, 62745, 64201, 65281, 65793, 66197, 69345, 69921, 73801, 74023, 74665, 75361, 77161, 80581, 85281, 85489, 87061, 87249, 88357, 90751, 92929, 94657, 95281, 96321, 97921, ...
9	91, 121, 671, 703, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601, 7381, 8401, 8911, 10585, 11011, 12403, 14383, 15203, 15457, 15841, 16471, 16531, 18721, 19345, 23521, 24661, 24727, 28009, 29161, 29341, 30857, 31621, 31697, 32791, 38503, 41041, 44287, 46657, 46999, 47197, 49051, 49141, 50881, 52633, 53131, 55261, 55969, 63139, 63973, 65485, 68887, 72041, 74593, 75361, 76627, 79003, 82513, 83333, 83665, 87913, 88561, 88573, 88831, 90751, 93961, 96139, 97567, ...
10	9, 91, 481, 1729, 4187, 6533, 6601, 8149, 8401, 10001, 11111, 11169, 11649, 12801, 15841, 19201, 20961, 21931, 24013, 34441, 41041, 50851, 50881, 63973, 69921, 75361, 79003, 83119, 94139, 95161, 97681, ...
11	133, 793, 2047, 2465, 4577, 4921, 5041, 5185, 12403, 13333, 14521, 15841, 17711, 18705, 23377, 34945, 43213, 43739, 47611, 48283, 49105, 49141, 49601, 50737, 50997, 55537, 56057, 57929, 58969, 62745, 68137, 74089, 85879, 86347, 87913, 88831, 94945, ...
12	91, 133, 145, 247, 385, 1649, 1729, 2041, 2233, 2821, 3553, 8911, 9073, 10585, 12673, 13051, 13333, 13345, 13585, 14905, 15841, 16471, 18721, 19517, 20737, 20881, 21361, 24013, 24727, 25681, 26467, 26785, 27985, 29341, 29539, 30745, 31483, 31621, 33553, 34219, 34861, 35881, 37345, 38311, 38503, 38665, 40321, 41041, 46657, 49141, 52633, 53083, 59185, 61309, 63973, 65569, 66637, 67861, 74305, 75361, 78793, 79381, 79501, 80185, 87841, 88705, 88831, 89089, 93961, 97351, ...
13	85, 105, 1099, 1785, 5149, 7107, 8841, 8911, 9577, 9637, 13019, 14491, 15505, 17803, 19757, 20881, 22177, 23521, 26521, 30073, 30889, 35371, 44173, 45629, 49105, 54097, 56033, 57205, 70801, 75241, 82733, 83333, 85285, 86347, 87681, 91001, ...
14	15, 65, 793, 841, 2465, 2743, 3277, 5713, 6541, 7171, 7449, 7585, 9073, 12545, 15457, 18721, 21667, 22261, 23521, 34441, 38221, 38417, 40385, 40501, 41371, 46657, 49471, 58255, 68401, 71969, 79003, 88381, 90241, 91681, 95033, 96049, 97469, ...
15	1687, 1729, 1921, 3277, 6541, 14041, 14701, 15409, 15841, 19201, 25313, 31021, 41041, 47461, 49241, 50401, 54241, 54649, 58969, 60691, 67249, 73801, 75361, 82733, 88831, 97921, ...
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, 10261, 10585, 11305, 12403, 12801, 13019, 13741, 13747, 13981, 14351, 14491, 14701, 15051, 15211, 15709, 15841, 16021, 16471, 16705, 18705, 18721, 19669, 19951, 20191, 20485, 23001, 23377, 24727, 25351, 25761, 26335, 26599, 27511, 28645, 29341, 30121, 30889, 31417, 31609, 31621, 33153, 33227, 33355, 34945, 35333, 38503, 39865, 40501, 40951, 41041, 41665, 42121, 42127, 42799, 45551, 45991, 46513, 46657, 47197, 47611, 48599, 49141, 49155, 49981, 50737, 51319, 52633, 53131, 55245, 57421, 60701, 60787, 61447, 62745, 63973, 64821, 65077, 65281, 68101, 68251, 72631, 72885, 73555, 74563, 74665, 75361, 76627, 76921, 77879, 78013, 79003, 80581, 81631, 81915, 83333, 83665, 85489, 87249, 88357, 88561, 88831, 90751, 91001, 92701, 93961, 98671, ...
17	9, 91, 145, 781, 1111, 1305, 2821, 4033, 4187, 5365, 5833, 6697, 7171, 12673, 15805, 19345, 19729, 21781, 22791, 24211, 26245, 31621, 33001, 33227, 34441, 35371, 38081, 42127, 46657, 49771, 62745, 71071, 74665, 77293, 78881, 80185, 88831, 93961, 96433, 97921, 98671, ...
18	25, 49, 65, 325, 343, 425, 1105, 1225, 1369, 1387, 1729, 1921, 2465, 2977, 4577, 5725, 5833, 5941, 6305, 6601, 7345, 10585, 11305, 11425, 12025, 15505, 15793, 15841, 18631, 19465, 22393, 22411, 27937, 28153, 29341, 30457, 30889, 35425, 39817, 39865, 40501, 41041, 41159, 43225, 46657, 49141, 50737, 52633, 54145, 60385, 60685, 60691, 74425, 75361, 80137, 84721, 89425, 90113, 90751, 91001, 94129, 99451, 99937, ...
19	9, 45, 49, 169, 343, 1849, 2353, 2701, 3201, 4033, 4681, 6541, 6697, 7957, 8281, 9997, 12403, 13213, 13747, 13833, 15251, 16531, 18769, 19201, 19729, 24761, 30589, 31621, 31861, 32477, 34945, 37681, 41003, 41041, 47593, 49141, 49771, 59585, 63139, 64681, 65161, 66421, 68257, 73555, 75361, 96049, ...
20	21, 671, 889, 1281, 1729, 1891, 2059, 2761, 3201, 5461, 6601, 7999, 12801, 13051, 15311, 15841, 16441, 21667, 25681, 31369, 34861, 35169, 37901, 38989, 41041, 42127, 49771, 50737, 52521, 54811, 57981, 64681, 68251, 75361, 78961, 85591, 86241, 88831, 89281, 92509, 93031, 96049, 97921, ...
21	221, 703, 793, 1045, 3781, 7363, 9061, 10945, 11647, 13051, 17767, 19345, 19669, 19909, 22681, 27133, 30073, 30745, 31021, 35785, 38503, 38665, 41353, 43213, 46657, 58829, 79081, 80137, 83569, 85285, 88357, 92509, 96049, ...
22	21, 91, 169, 345, 485, 1183, 1247, 2047, 2465, 5551, 7665, 10465, 11557, 14111, 15229, 15841, 16393, 17169, 17767, 18705, 19909, 20881, 21667, 23651, 31417, 33465, 34945, 38503, 47197, 49141, 53131, 62745, 64907, 70579, 72581, 76921, 88705, 89281, 90851, 98385, 99541, ...
23	169, 265, 553, 1271, 1729, 2465, 2701, 4033, 4371, 4681, 6533, 6541, 7189, 7957, 8321, 8651, 8911, 9805, 11713, 14905, 14981, 18721, 19513, 19517, 20801, 25201, 28897, 31861, 34133, 38665, 41041, 44173, 44785, 46657, 47611, 47783, 50737, 52633, 57401, 62849, 75361, 80401, 82513, 86101, 93457, 96049, ...
24	25, 175, 553, 949, 1541, 1729, 1825, 1975, 2701, 4537, 6931, 7501, 9361, 10465, 10585, 12025, 13825, 14425, 15025, 15841, 19345, 19513, 21349, 25201, 25273, 25477, 29185, 29341, 29665, 35425, 38323, 40369, 41041, 42121, 42127, 43873, 46657, 47617, 49141, 50881, 52417, 52633, 55969, 63701, 75361, 80137, 83333, 83665, 85609, 88561, 89281, 94753, ...
25	217, 561, 781, 1541, 1729, 1891, 2821, 4123, 5461, 5611, 5731, 6601, 7449, 7813, 8029, 8911, 9881, 11041, 12801, 13021, 13333, 13981, 14981, 15751, 15841, 16297, 17767, 21361, 22791, 23653, 24211, 25327, 25351, 29341, 29539, 30673, 32021, 35371, 36661, 36991, 38081, 40501, 41041, 42127, 44173, 44801, 45141, 46657, 47641, 48133, 50737, 50997, 52633, 53083, 53971, 56033, 58807, 63973, 67921, 68101, 68251, 75361, 79381, 88831, 90241, 98173, ...
26	9, 25, 27, 45, 217, 225, 475, 703, 925, 1065, 3825, 5041, 5425, 8029, 9073, 11005, 11041, 13833, 16725, 17575, 23653, 24073, 24727, 25299, 25425, 31651, 33227, 35881, 47197, 55537, 60701, 61975, 63025, 68251, 70561, 76627, 79003, 85321, 91525, 93961, 95051, 96641, ...
27	121, 133, 259, 365, 481, 703, 1649, 1729, 1891, 2821, 3281, 4033, 4921, 5461, 7381, 7585, 8401, 8911, 9809, 9841, 9881, 10585, 11041, 12403, 13019, 13073, 13333, 15457, 15841, 16021, 16531, 18721, 19345, 21901, 23521, 24661, 26467, 26599, 27265, 28009, 29341, 29891, 31609, 31621, 35113, 35371, 37969, 41041, 44287, 46657, 47197, 48133, 49141, 50171, 50881, 52633, 52801, 55537, 55969, 58201, 59641, 63139, 63973, 69469, 74023, 74593, 75361, 79003, 79381, 82513, 82853, 87061, 87913, 88573, 92509, 92833, 92929, 93961, 94657, 97567, 97921, 98881, ...
28	9, 27, 261, 361, 529, 785, 1431, 2041, 2465, 3201, 3277, 4699, 5149, 7065, 8401, 13357, 13833, 14981, 17767, 27133, 28009, 31753, 32551, 33227, 35443, 36801, 38503, 43213, 46657, 50737, 58969, 68143, 80137, 81317, 82513, 88705, 96139, 97567, 97831, ...
29	15, 91, 341, 469, 871, 2257, 4371, 4411, 5149, 5185, 6097, 8401, 8841, 11581, 12431, 15577, 15841, 16471, 19093, 22281, 25681, 27613, 28009, 29539, 31417, 33001, 41041, 46657, 48133, 49141, 54913, 57889, 79003, 98301, ...
30	49, 133, 341, 403, 637, 871, 901, 931, 1729, 2059, 2077, 3277, 4081, 4097, 6031, 6409, 8023, 8401, 9881, 11041, 11809, 15841, 17593, 24929, 26599, 27001, 27133, 30889, 33227, 38503, 41041, 42127, 43213, 48133, 50881, 52801, 54961, 56033, 57137, 63973, 65569, 66197, 75361, 79003, 92929, 96049, ...

關於n的最小歐拉－雅可比偽素數

n	最小歐拉－雅可比偽素數	n	最小歐拉－雅可比偽素數	n	最小歐拉－雅可比偽素數	n	最小歐拉－雅可比偽素數
1	9	33	545	65	33	97	49
2	561	34	33	66	65	98	9
3	121	35	9	67	33	99	25
4	341	36	35	68	25	100	9
5	781	37	9	69	35	101	25
6	217	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	91	44	9	76	15	108	91
13	85	45	481	77	39	109	9
14	15	46	9	78	77	110	111
15	1687	47	65	79	39	111	55
16	15	48	49	80	9	112	65
17	9	49	25	81	91	113	57
18	25	50	49	82	9	114	115
19	9	51	25	83	21	115	57
20	21	52	51	84	85	116	9
21	221	53	9	85	21	117	49
22	21	54	55	86	85	118	9
23	169	55	9	87	247	119	15
24	25	56	55	88	87	120	91
25	217	57	25	89	9	121	15
26	9	58	57	90	91	122	65
27	121	59	15	91	9	123	85
28	9	60	481	92	91	124	25
29	15	61	15	93	25	125	9
30	49	62	9	94	93	126	25
31	15	63	529	95	1891	127	9
32	25	64	9	96	95	128	49

參見

• 費馬偽素數
• 卡邁克爾數
• 歐拉偽素數
• 強偽質數

參考文獻

1. Carl Pomerance; John L. Selfridge; Samuel S. Wagstaff, Jr. The pseudoprimes to 25·10⁹ (PDF). Mathematics of Computation. July 1980, 35 (151): 1003–1026 [2016-09-22]. doi:10.1090/S0025-5718-1980-0572872-7. （原始內容存檔 (PDF)於2016-12-03）.