# Difference between revisions of "Pseudo-prime"

(Category:Number theory) |
(mention Fermat's little theorem) |
||

Line 1: | Line 1: | ||

{{TEX|done}} | {{TEX|done}} | ||

− | Traditionally, a composite natural number $n$ is called a pseudo-prime if $2^{n-1} \equiv 1$ modulo $n$, 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 $n$, the first five being | + | Traditionally, a composite natural number $n$ is called a pseudo-prime if $2^{n-1} \equiv 1$ modulo $n$, for it has long been known that primes have this property: this is [[Fermat's little theorem]]. (The term is apparently due to D.H. Lehmer.) There are infinitely many such $n$, the first five being |

$$ | $$ | ||

341,\,561,\,645,\,1105,\,1387\ . | 341,\,561,\,645,\,1105,\,1387\ . | ||

$$ | $$ | ||

− | More recently, the concept has been extended to include any composite number that acts like a prime in some realization of a [[ | + | 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-$b$ pseudo-primes, satisfying $b^{n-1}\equiv1$ modulo $n$; | 1) ordinary base-$b$ pseudo-primes, satisfying $b^{n-1}\equiv1$ modulo $n$; | ||

− | 2) Euler base-$b$ pseudo-primes, whose [[ | + | 2) Euler base-$b$ pseudo-primes, whose [[Jacobi symbol]] with $b$ satisfies |

$$ | $$ | ||

b^{(n-1)/2} \equiv\left({\frac{b}{n}}\right) = \pm 1 | b^{(n-1)/2} \equiv\left({\frac{b}{n}}\right) = \pm 1 | ||

Line 20: | Line 20: | ||

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

− | The concept of a pseudo-prime has been generalized to include primality tests based on finite fields and elliptic curves (cf. also [[ | + | 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 [[#References|[a3]]], [[#References|[a5]]]. |

The complementary concept is also of interest. The base $b$ is called a (Fermat) witness for $n$ if $n$ is composite and not a base-$b$ pseudo-prime. Euler and strong witnesses are similarly defined. If $W(n)$, the smallest strong witness for $n$, grows sufficiently slowly, there is a polynomial-time algorithm for primality. It is known that $W(n)$ is not bounded [[#References|[a2]]], but if an extended version of the Riemann hypothesis (cf. [[Riemann hypotheses|Riemann hypotheses]]) holds, then $W(n) \le 2(\log n)^2$ [[#References|[a1]]]. | The complementary concept is also of interest. The base $b$ is called a (Fermat) witness for $n$ if $n$ is composite and not a base-$b$ pseudo-prime. Euler and strong witnesses are similarly defined. If $W(n)$, the smallest strong witness for $n$, grows sufficiently slowly, there is a polynomial-time algorithm for primality. It is known that $W(n)$ is not bounded [[#References|[a2]]], but if an extended version of the Riemann hypothesis (cf. [[Riemann hypotheses|Riemann hypotheses]]) holds, then $W(n) \le 2(\log n)^2$ [[#References|[a1]]]. | ||

Line 29: | Line 29: | ||

<TR><TD valign="top">[a2]</TD> <TD valign="top"> 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</TD></TR> | <TR><TD valign="top">[a2]</TD> <TD valign="top"> 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</TD></TR> | ||

<TR><TD valign="top">[a3]</TD> <TD valign="top"> F. Morain, "Pseudoprimes: a survey of recent results" , ''Proc. Eurocode '92'' , Springer (1993) pp. 207–215</TD></TR> | <TR><TD valign="top">[a3]</TD> <TD valign="top"> F. Morain, "Pseudoprimes: a survey of recent results" , ''Proc. Eurocode '92'' , Springer (1993) pp. 207–215</TD></TR> | ||

− | <TR><TD valign="top">[a4]</TD> <TD valign="top"> C. Pomerance, J.L. Selfridge, S.S. Wagstaff, Jr., "The pseudoprimes to $25\cdot10^9$" ''Math. Comp.'' , '''35''' (1980) pp. 1003–1026</TD></TR> | + | <TR><TD valign="top">[a4]</TD> <TD valign="top"> C. Pomerance, J.L. Selfridge, S.S. Wagstaff, Jr., "The pseudoprimes to $25\cdot10^9$" ''Math. Comp.'' , '''35''' (1980) pp. 1003–1026. Zbl 0444.10007. DOI 10.2307/2006210</TD></TR> |

<TR><TD valign="top">[a5]</TD> <TD valign="top"> P. Ribenboim, "The book of prime number records" , Springer (1989) (Edition: Second)</TD></TR> | <TR><TD valign="top">[a5]</TD> <TD valign="top"> P. Ribenboim, "The book of prime number records" , Springer (1989) (Edition: Second)</TD></TR> | ||

<TR><TD valign="top">[a6]</TD> <TD valign="top"> N.J.A. Sloane, S. Plouffe, "The encyclopedia of integer sequences" , Acad. Press (1995)</TD></TR> | <TR><TD valign="top">[a6]</TD> <TD valign="top"> N.J.A. Sloane, S. Plouffe, "The encyclopedia of integer sequences" , Acad. Press (1995)</TD></TR> |

## Latest revision as of 17:58, 8 November 2014

Traditionally, a composite natural number $n$ is called a pseudo-prime if $2^{n-1} \equiv 1$ modulo $n$, for it has long been known that primes have this property: this is Fermat's little theorem. (The term is apparently due to D.H. Lehmer.) There are infinitely many such $n$, the first five being $$ 341,\,561,\,645,\,1105,\,1387\ . $$

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-$b$ pseudo-primes, satisfying $b^{n-1}\equiv1$ modulo $n$;

2) Euler base-$b$ pseudo-primes, whose Jacobi symbol with $b$ satisfies $$ b^{(n-1)/2} \equiv\left({\frac{b}{n}}\right) = \pm 1 $$

3) strong base-$b$ pseudo-primes, for which the sequence $b^{s.2^i}$ modulo $n$ for $i=0,\ldots, r$ is either always $1$, or contains $-1$. (Here $n-1 = 2^r.s$ with $s$ odd.)

For each $b$, the implications 3)$\Rightarrow$2)$\Rightarrow$1) hold. A number $n$ that is an ordinary base-$b$ pseudo-prime for all $b$ prime to $n$ 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 $b$ is called a (Fermat) witness for $n$ if $n$ is composite and not a base-$b$ pseudo-prime. Euler and strong witnesses are similarly defined. If $W(n)$, the smallest strong witness for $n$, grows sufficiently slowly, there is a polynomial-time algorithm for primality. It is known that $W(n)$ is not bounded [a2], but if an extended version of the Riemann hypothesis (cf. Riemann hypotheses) holds, then $W(n) \le 2(\log n)^2$ [a1].

#### References

[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 $25\cdot10^9$" Math. Comp. , 35 (1980) pp. 1003–1026. Zbl 0444.10007. DOI 10.2307/2006210 |

[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: http://encyclopediaofmath.org/index.php?title=Pseudo-prime&oldid=33851