Adleman–Pomerance–Rumely primality test

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In computational number theory, the Adleman–Pomerance–Rumely primality test is an algorithm for determining whether a number is prime. Unlike other, more efficient algorithms for this purpose, it avoids the use of random numbers, so it is a deterministic primality test. It is named after its discoverers, Leonard Adleman, Carl Pomerance, and Robert Rumely. The test involves arithmetic in cyclotomic fields.

It was later improved by Henri Cohen and Hendrik Willem Lenstra and called APRT-CL (or APRCL). It is often used with UBASIC under the name APRT-CLE (APRT-CL extended) and can test primality of an integer n in time:

$O((\ln n)^{{c\,\ln \,\ln \,\ln n}}).$

External links

APR and APR-CL
A factoring applet that uses APR-CL on certain conditions (source code included)
Pari/GP uses APR-CL conditionally in its implementation of isprime().

References

Adleman, Leonard M.; Pomerance, Carl; Rumely, Robert S. (1983). "On distinguishing prime numbers from composite numbers". Annals of Mathematics 117 (1): 173–206. doi:10.2307/2006975.
Cohen, Henri; Lenstra, Hendrik W., Jr. (1984). "Primality testing and Jacobi sums". Mathematics of Computation 42 (165): 297–330. doi:10.2307/2007581.
Riesel, Hans (1994). Prime Numbers and Computer Methods for Factorization. Birkhauser. pp. 131–136. ISBN 0-8176-3743-5.

v t e Number-theoretic algorithms

Primality tests	AKS test APR test Baillie–PSW ECPP test Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius test Solovay–Strassen Miller–Rabin Trial division

Prime-generating	Sieve of Atkin Sieve of Eratosthenes Sieve of Sundaram Wheel factorization

Integer factorization	Continued fraction (CFRAC) Dixon's Lenstra elliptic curve (ECM) Euler's Pollard's rho p − 1 p + 1 Quadratic sieve (QS) General number field sieve (GNFS) Special number field sieve (SNFS) Rational sieve Fermat's Shanks' square forms Trial division Shor's

Multiplication	Ancient Egyptian Karatsuba Toom–Cook Schönhage–Strassen Fürer's

Discrete logarithm	Baby-step giant-step Pollard rho Pollard kangaroo Pohlig–Hellman Index calculus Function field sieve

Greatest common divisor	Binary Euclidean Extended Euclidean Lehmer's

Modular square root	Cipolla Pocklington's Tonelli–Shanks

Other algorithms	Chakravala Cornacchia Integer relation Integer square root Modular exponentiation Schoof's

Italics indicate that algorithm is for numbers of special forms Smallcaps indicate a deterministic algorithm

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