# Euler–Jacobi pseudoprime

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In number theory, an odd integer n is called an Euler–Jacobi probable prime (or, more commonly, an Euler probable prime) to base a, if a and n are coprime, and

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

where $\left(\frac{a}{n}\right)$ is the Jacobi symbol.

If n is a composite integer that satisfies the above congruence, then n is called an Euler–Jacobi pseudoprime (or, more commonly, an Euler pseudoprime).

## 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.

The smallest Euler–Jacobi pseudoprime base 2 is 561. There are 11347 Euler–Jacobi pseudoprimes base 2 that are less than 25·109 (see ) (page 1005 of [1]).

In the literature (for example,[1]), an Euler–Jacobi pseudoprime as defined above is often called simply an Euler pseudoprime.

## Examples

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

 a Euler–Jacobi pseudoprimes in base 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