Euler-Jacobi pseudoprime

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.

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.

It should be noted that these numbers are, in some sources, called Euler pseudoprimes.

The table below gives all Euler-Jacobi pseudoprimes less than 10000 for some prime bases a, this table is in the process of being checked and should be used with caution until this notice is removed.

a
2	561, 1105, 1729, 1905, 2047, 2465, 3277, 4033, 4681, 6601, 8321, 8481
3	121, 1729, 2821, 7381, 8401
5	781, 1541, 1729, 5461, 5611, 6601, 7449
7	25, 703, 2101, 2353, 2465, 3277
11	133, 793, 2465, 4577, 4921, 5041, 5185
13	105, 1785, 5149, 7107, 8841, 9577, 9637
17	9, 145, 781, 1305, 2821, 4033, 5833, 6697
19	9, 45, 49, 169, 1849, 2353, 3201, 4033, 4681, 6541, 6697, 8281
23	169, 265, 553, 1729, 2465, 4033, 4681, 6533, 6541, 7189, 8321, 8911
29	91, 341, 469, 871, 2257, 5149, 5185, 6097, 8401, 8841
31	15, 49, 133, 481, 2465, 6241, 7449, 9131
37	9, 451, 469, 589, 817, 1233, 1333, 1729, 3781, 3913, 4521, 5073, 8905, 9271
41	21, 105, 841, 1065, 1281, 1417, 2465, 2701, 3829, 8321
43	21, 25, 33, 77, 105, 185, 385, 481, 561, 777, 825, 973, 1105, 1541, 1729,  1825, 2465, 2553, 2821, 2849, 3281, 3439, 3781, 4033, 4417, 6105, 6369, 6545, 6601, 6697, 7825,
47	65, 69, 341, 345, 481, 561, 703, 721, 793, 897, 1105, 1649, 1729, 1891, 2257, 2465, 2737, 3145, 3201, 5185, 5461, 5865, 6305, 9361
53	9, 65, 91, 117, 561, 585, 1105, 1441, 1541, 1729, 2209, 2465, 2529, 2821, 2863, 3097, 3367, 3481, 3861, 5317, 5833, 6031, 6433, 9409
59	145, 451, 561, 645, 1105, 1141, 1247, 1541, 1661, 1729, 1991, 2413, 2465, 3097, 4681, 5611, 5729, 6191, 6533, 6601, 7421, 8149, 8321, 8705, 9637
61	15, 93, 217, 341, 465, 1261, 1441, 1729, 2465, 2821, 3565, 3661, 4061, 4577, 5461, 6541, 6601, 6697, 7613, 7905, 9305, 9937
67	33, 49, 217, 385, 561, 1105, 1309, 1519, 1705, 1729, 2209, 2465, 3201, 5797, 7633, 7701,  8029, 8321, 8371, 9073

See also:

• Probable prime