Adleman–Pomerance–Rumely primality test

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, commonly referred to as APR-CL. It can test primality of an integer n in time:

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

Software implementations

UBASIC provides an implementation under the name APRT-CLE (APR Test CL extended)
A factoring applet that uses APR-CL on certain conditions (source code included)
Pari/GP uses APR-CL conditionally in its implementation of isprime().
mpz_aprcl is an open source implementation using C and GMP.

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. Birkhäuser. pp. 131–136. ISBN 0-8176-3743-5.
APR and APR-CL

Number-theoretic algorithms
Primality tests	AKS test APR test Baillie–PSW Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius test Solovay–Strassen Miller–Rabin
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 Long 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 LLL Integer square root Modular exponentiation Schoof's
Italics indicate that algorithm is for numbers of special forms

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