In number theory, a probable prime is a number that passes a primality test. A strong probable prime is a number that passes a strong version of a primality test. A strong pseudoprime is a composite number that passes a strong version of a primality test. All primes pass these tests, but a small fraction of composites also pass, making them "false primes".
Formally, a composite number n = d · 2s + 1 with d being odd is called a strong (Fermat) pseudoprime to a relatively prime base a when one of the following conditions holds:
(If a number n satisfies one of the above conditions and we don't yet know whether it is prime, it is more precise to refer to it as a strong probable prime to base a. But if we know that n is not prime, then one may use the term strong pseudoprime.)
The definition of a strong pseudoprime depends on the base used; different bases have different strong pseudoprimes. The definition is trivially met if a ≡ ±1 mod n so these trivial bases are often excluded.
Properties of strong pseudoprimes
A strong pseudoprime to base a is always an Euler-Jacobi pseudoprime, an Euler pseudoprime  and a Fermat pseudoprime to that base, but not all Euler and Fermat pseudoprimes are strong pseudoprimes. Carmichael numbers may be strong pseudoprimes to some bases—for example, 561 is a strong pseudoprime to base 50—but not to all bases.
A composite number n is a strong pseudoprime to at most one quarter of all bases below n; thus, there are no "strong Carmichael numbers", numbers that are strong pseudoprimes to all bases. Thus given a random base, the probability that a number is a strong pseudoprime to that base is less than 1/4, forming the basis of the widely used Miller-Rabin primality test. However, Arnault  gives a 397-digit composite number that is a strong pseudoprime to every base less than 307. One way to prevent such a number from wrongfully being declared probably prime is to combine a strong probable prime test with a Lucas probable prime test, as in the Baillie-PSW primality test.
There are infinitely many strong pseudoprimes to any base.
The first strong pseudoprimes to base 2 are
- 2047, 3277, 4033, 4681, 8321, 15841, 29341, 42799, 49141, 52633, 65281, 74665, 80581, 85489, 88357, 90751, ... (sequence A001262 in OEIS).
The first to base 3 are
- 121, 703, 1891, 3281, 8401, 8911, 10585, 12403, 16531, 18721, 19345, 23521, 31621, 44287, 47197, 55969, 63139, 74593, 79003, 82513, 87913, 88573, 97567, ... (sequence A020229 in OEIS).
The first to base 5 are
- 781, 1541, 5461, 5611, 7813, 13021, 14981, 15751, 24211, 25351, 29539, 38081, 40501, 44801, 53971, 79381, ... (sequence A020231 in OEIS).
By testing the above conditions to several bases, one gets somewhat more powerful primality tests than by using one base alone. For example, there are only 13 numbers less than 25·109 that are strong pseudoprimes to bases 2, 3, and 5 simultaneously. They are listed in Table 7 of. The smallest such number is 25326001. This means that, if n is less than 25326001 and n is a strong probable prime to bases 2, 3, and 5, then n is prime.
Carrying this further, 3825123056546413051 is the smallest number that is a strong pseudoprime to the 9 bases 2, 3, 5, 7, 11, 13, 17, 19, and 23. See  and . So, if n is less than 3825123056546413051 and n is a strong probable prime to these 9 bases, then n is prime.
- Guy, Pseudoprimes. Euler Pseudoprimes. Strong Pseudoprimes. §A12 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 27-30, 1994.
- Carl Pomerance; John L. Selfridge, Samuel S. Wagstaff, Jr. (July 1980). "The pseudoprimes to 25·109". Mathematics of Computation 35 (151): 1003–1026. doi:10.1090/S0025-5718-1980-0572872-7.
- Monier, Evaluation and Comparison of Two Efficient Probabilistic Primality Testing Algorithms. Theoretical Computer Science, 12 pp. 97-108, 1980.
- Rabin, Probabilistic Algorithm for Testing Primality. Journal of Number Theory, 12 pp. 128-138, 1980.
- F. Arnault (August 1995). "Constructing Carmichael Numbers Which Are Strong Pseudoprimes to Several Bases". Journal of Symbolic Computation 20 (2): 151–161.
- Zhenxiang Zhang; Min Tang (2003). "Finding Strong Pseudoprimes to Several Bases. II". Mathematics of Computation 72 (244): 2085–2097. doi:10.1090/S0025-5718-03-01545-X.
- Jiang, Yupeng; Deng, Yingpu (2012). "Strong pseudoprimes to the first 9 prime bases". arXiv:1207.0063v1 [math.NT].