Pseudoprime

A pseudoprime is a Composite number which passes a test or sequence of tests which fail for most Composite numbers.Unfortunately, some authors drop the ``Composite'' requirement, calling any number which passes the specified tests apseudoprime even if it is Prime. Pomerance, Selfridge, and Wagstaff (1980) restrict their use of ``pseudoprime'' to OddComposite numbers. ``Pseudoprime'' used without qualification means Fermat Pseudoprime.

Carmichael Numbers are Odd Composite numbers which are pseudoprimes to every base; they aresometimes called Absolute Pseudoprimes. The following table gives the number ofFermat Pseudoprimes psp, Euler Pseudoprimes epsp, andStrong Pseudoprimes spsp to the base 2, as well as Carmichael Numbers CN which are less the first few powers of 10 (Guy 1994).

psp(2)	3	22	78	245	750	2057	5597	14884
epsp(2)	1	12	36	114	375	1071	2939	7706
spsp(2)	0	5	16	46	162	488	1282	3291
CN	1	7	16	43	105	255	646	1547

References

Grantham, J. ``Frobenius Pseudoprimes.'' http://www.clark.net/pub/grantham/pseudo/pseudo1.ps

Grantham, J. ``Pseudoprimes/Probable Primes.'' http://www.clark.net/pub/grantham/pseudo.

Guy, R. K. ``Pseudoprimes. Euler Pseudoprimes. Strong Pseudoprimes.'' §A12 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 27-30, 1994.

Pomerance, C.; Selfridge, J. L.; and Wagstaff, S. S. ``The Pseudoprimes to .'' Math. Comput. 35, 1003-1026, 1980. Available electronically from ftp://sable.ox.ac.uk/pub/math/primes/ps2.Z.

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