Explicit primality criteria for

Authors:
Andreas Stein and H. C. Williams

Journal:
Math. Comp. **69** (2000), 1721-1734

MSC (1991):
Primary 11Y11; Secondary 11Y16

DOI:
https://doi.org/10.1090/S0025-5718-00-01212-6

Published electronically:
February 23, 2000

MathSciNet review:
1697651

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Deterministic polynomial time primality criteria for have been known since the work of Lucas in 1876-1878. Little is known, however, about the existence of deterministic polynomial time primality tests for numbers of the more general form , where is any fixed prime. When we show that it is always possible to produce a Lucas-like deterministic test for the primality of which requires that only modular multiplications be performed modulo , as long as we can find a prime of the form such that is not divisible by . We also show that for all with such a can be found very readily, and that the most difficult case in which to find a appears, somewhat surprisingly, to be that for . Some explanation is provided as to why this case is so difficult.

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Additional Information

**Andreas Stein**

Affiliation:
University of Waterloo, Department of Combinatorics and Optimization, Waterloo, Ontario, Canada N2L 3G1

Email:
astein@cacr.math.uwaterloo.ca

**H. C. Williams**

Affiliation:
University of Manitoba, Department of Computer Science, Winnipeg, Manitoba, Canada R3T 2N2

Email:
williams@cs.umanitoba.ca

DOI:
https://doi.org/10.1090/S0025-5718-00-01212-6

Keywords:
Primality test,
Mersenne numbers,
Lucas functions,
Gauss sums,
covering sets

Received by editor(s):
October 24, 1997

Received by editor(s) in revised form:
October 23, 1998

Published electronically:
February 23, 2000

Additional Notes:
Research supported by NSERC of Canada Grant $#A7649$.

Article copyright:
© Copyright 2000
American Mathematical Society