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Traditionally, a composite natural number is called a pseudo-prime if modulo , for it has long been known that primes have this property. (The term is apparently due to D.H. Lehmer.) There are infinitely many such , the first five being

More recently, the concept has been extended to include any composite number that acts like a prime in some realization of a probabilistic primality test. That is, it satisfies some easily computable necessary, but not sufficient, condition for primality. Pseudo-primes in this larger sense include:

1) ordinary base- pseudo-primes, satisfying modulo ;

2) Euler base- pseudo-primes, whose Jacobi symbol with satisfies

3) strong base- pseudo-primes, for which the sequence modulo , , is either always , or contains . (Here with odd.)

For each , the implications 3)2)1) hold. A number that is an ordinary base- pseudo-prime for all prime to is called a Carmichael number. Analogous numbers for the other two categories do not exist.

For a thorough empirical study of pseudo-primes, see [a4]. Lists of pseudo-primes to various small bases can be found in [a6].

The concept of a pseudo-prime has been generalized to include primality tests based on finite fields and elliptic curves (cf. also Finite field; Elliptic curve). For reviews of this work, see [a3], [a5].

The complementary concept is also of interest. The base is called a (Fermat) witness for if is composite but not a base- pseudo-prime. Euler and strong witnesses are similarly defined. If , the smallest strong witness for , grows sufficiently slowly, there is a polynomial-time algorithm for primality. It is known that is not bounded [a2], but if an extended version of the Riemann hypothesis (cf. Riemann hypotheses) holds, then [a1].


[a1] E. Bach, "Analytic methods in the analysis and design of number-theoretic algorithms" , MIT (1985)
[a2] W.R. Alford, A. Granville, C. Pomerance, "On the difficulty of finding reliable witnesses" , Algorithmic Number Theory, First Internat. Symp., ANTS-I , Lecture Notes in Computer Science , 877 , Springer (1994) pp. 1–16
[a3] F. Morain, "Pseudoprimes: a survey of recent results" , Proc. Eurocode '92 , Springer (1993) pp. 207–215
[a4] C. Pomerance, J.L. Selfridge, S.S. Wagstaff, Jr., "The pseudoprimes to " Math. Comp. , 35 (1980) pp. 1003–1026
[a5] P. Ribenboim, "The book of prime number records" , Springer (1989) (Edition: Second)
[a6] N.J.A. Sloane, S. Plouffe, "The encyclopedia of integer sequences" , Acad. Press (1995)
How to Cite This Entry:
Pseudo-prime. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E. Bach (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article