Euler–Jacobi pseudoprime

In number theory, an odd composite integer n is called an Euler–Jacobi pseudoprime to base a, if a and n are coprime, and

a^{(n − 1)/2} ≡ (a/n) (mod n),

where (a/n) is the Jacobi symbol.

Properties

The motivation for this definition is the fact that all prime numbers n satisfy the above equation, as explained in the Legendre symbol article. The equation can be tested rather quickly, which can be used for probabilistic primality testing. These tests are over twice as strong as tests based on Fermat's little theorem.

Every Euler–Jacobi pseudoprime is also a Fermat pseudoprime and an Euler pseudoprime. There are no numbers which are Euler–Jacobi pseudoprimes to all bases as Carmichael numbers are. Solovay and Strassen showed that for every composite n, for at least n/2 bases less than n, n is not an Euler–Jacobi pseudoprime.

Examples

The table below gives all Euler–Jacobi pseudoprimes less than 10000 for some prime bases a.

a
2	561, 1105, 1729, 1905, 2047, 2465, 3277, 4033, 4681, 6601, 8321, 8481
3	121, 703, 1729, 1891, 2821, 3281, 7381, 8401, 8911
5	781, 1541, 1729, 5461, 5611, 6601, 7449, 7813
7	25, 325, 703, 2101, 2353, 2465, 3277, 4525
11	133, 793, 2047, 2465, 4577, 4921, 5041, 5185
13	85, 105, 1099, 1785, 5149, 7107, 8841, 8911, 9577, 9637
17	9, 91, 145, 781, 1111, 1305, 2821, 4033, 4187, 5365, 5833, 6697, 7171
19	9, 45, 49, 169, 343, 1849, 2353, 2701, 3201, 4033, 4681, 6541, 6697, 7957, 8281, 9997
23	169, 265, 553, 1271, 1729, 2465, 2701, 4033, 4371, 4681, 6533, 6541, 7189, 7957, 8321, 8651, 8911, 9805
29	15, 91, 341, 469, 871, 2257, 4371, 4411, 5149, 5185, 6097, 8401, 8841
31	15, 49, 133, 481, 931, 2465, 6241, 7449, 8911, 9131
37	9, 451, 469, 589, 685, 817, 1233, 1333, 1729, 3781, 3913, 4521, 5073, 8905, 9271
41	21, 105, 231, 671, 703, 841, 1065, 1281, 1387, 1417, 2465, 2701, 3829, 8321, 8911
43	21, 25, 185, 385, 925, 1541, 1729, 1807, 2465, 2553, 2849, 3281, 3439, 3781, 4417, 6545, 7081, 8857
47	65, 85, 221, 341, 345, 703, 721, 897, 1105, 1649, 1729, 1891, 2257, 2465, 5461, 5865, 6305, 9361, 9881
53	9, 27, 91, 117, 1405, 1441, 1541, 2209, 2529, 2863, 3367, 3481, 5317, 6031, 9409
59	15, 145, 451, 1141, 1247, 1541, 1661, 1991, 2413, 2465, 3097, 4681, 5611, 6191, 7421, 8149, 9637
61	15, 217, 341, 1261, 2465, 2701, 2821, 3565, 3661, 6541, 6601, 6697, 7613, 7905
67	33, 49, 217, 561, 703, 1105, 1309, 1519, 1729, 2209, 2245, 5797, 6119, 7633, 8029, 8371
71	9, 35, 45, 1387, 1729, 1921, 2071, 2209, 2321, 2701, 4033, 6541, 7957, 8365, 8695, 9809
73	9, 65, 205, 259, 333, 369, 533, 585, 1441, 1729, 1921, 2553, 2665, 3439, 5257, 6697
79	39, 49, 65, 91, 301, 559, 637, 1649, 2107, 2465, 2701, 3913, 6305, 6533, 7051, 8321, 9881
83	21, 65, 231, 265, 561, 689, 703, 861, 1105, 1241, 1729, 2665, 3277, 3445, 4411, 5713, 6601, 6973, 7665, 8421
89	9, 15, 45, 153, 169, 1035, 1441, 2097, 2611, 2977, 3961, 4187, 5461, 6697, 7107, 7601, 7711
97	49, 105, 341, 469, 481, 949, 973, 1065, 2701, 3283, 3577, 4187, 4371, 4705, 6811, 8023, 8119, 8911, 9313

See also

• Probable prime