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# 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 and called APRT-CL (or APRCL). It can test primality of an integer n in time:

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

Software implementations

UBASIC provides an implementation under the name APRT-CLE (APRT-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().

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.

APR and APR-CL

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