pseudoprime test

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1: 27.22 Software

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Maple. isprime combines a strong pseudoprime test and a Lucas pseudoprime test. ifactor uses cfrac (§27.19) after exhausting trial division. Brent–Pollard rho, Square Forms Factorization, and ecm are available also; see §27.19.

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Mathematica. PrimeQ combines strong pseudoprime tests for the bases 2 and 3 and a Lucas pseudoprime test. No known composite numbers pass these three tests, and Bleichenbacher (1996) has shown that this combination of tests proves primality for integers below $10^{16}$ . Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard $p-1$ , and then cfrac after trial division. See §27.19. ecm is available also, and the Multiple Polynomial Quadratic sieve is expected in a future release.

For additional Mathematica routines for factorization and primality testing, including several different pseudoprime tests, see Bressoud and Wagon (2000).

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ECMNET Project. Links to software for elliptic curve methods of factorization and primality testing.

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Number Theory Web. References and links to software for factorization and primality testing.

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Wolfram’s Mathworld. Descriptions, references, and Mathematica algorithms for factorization and primality testing.

2: 27.12 Asymptotic Formulas: Primes

… ►For current records see The Great Internet Mersenne Prime Search. ►A pseudoprime test is a test that correctly identifies most composite numbers. …Descriptions and comparisons of pseudoprime tests are given in Bressoud and Wagon (2000, §§2.4, 4.2, and 8.2) and Crandall and Pomerance (2005, §§3.4–3.6). …

3: 4.48 Software

… ►Links to research literature for the Lambert

W

-function and for test software are included also. … ►

§4.48(v) Testing

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4: 27.18 Methods of Computation: Primes

… ►These algorithms are used for testing primality of Mersenne numbers,

2^{n}-1

, and Fermat numbers,

2^{2^{n}}+1

. … ►The APR (Adleman–Pomerance–Rumely) algorithm for primality testing is based on Jacobi sums. … ►The AKS (Agrawal–Kayal–Saxena) algorithm is the first deterministic, polynomial-time, primality test. …

5: 1.16 Distributions

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§1.16(i) Test Functions

… ►A test function is an infinitely differentiable function of compact support. … ► ►The linear space of all test functions with the above definition of convergence is called a test function space. … …

6: 35.9 Applications

… ►For other statistical applications of

{{}_{p}F_{q}}

functions of matrix argument see Perlman and Olkin (1980), Groeneboom and Truax (2000), Bhaumik and Sarkar (2002), Richards (2004) (monotonicity of power functions of multivariate statistical test criteria), Bingham et al. (1992) (Procrustes analysis), and Phillips (1986) (exact distributions of statistical test criteria). …

7: David M. Bressoud

… ► 227, in 1980, Factorization and Primality Testing, published by Springer-Verlag in 1989, Second Year Calculus from Celestial Mechanics to Special Relativity, published by Springer-Verlag in 1992, A Radical Approach to Real Analysis, published by the Mathematical Association of America in 1994, with a second edition in 2007, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, published by the Mathematical Association of America and Cambridge University Press in 1999, A Course in Computational Number Theory (with S. …

8: 1.1 Special Notation

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$x, y$	real variables.
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$\phi$	a testing function.
$\left\langle\Lambda,\phi\right\rangle$	action of distribution $\Lambda$ on test function $\phi$ .
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9: Daniel W. Lozier

… ►His research interests have centered on numerical analysis, special functions, computer arithmetic, and mathematical software construction and testing. …

10: 23.20 Mathematical Applications

… ►The resulting points are then tested for finite order as follows. … ►

§23.20(iii) Factorization

►§27.16 describes the use of primality testing and factorization in cryptography. …