“Lucas-Lehmer Test”的概念、定义、习题解题方法及翻译-数学辞典

单词

Lucas-Lehmer Test

释义

Lucas-Lehmer Test

A Mersenne Number is prime Iff divides , where and

(1)

for

. The first few terms of this series are 4, 14, 194, 37634, 1416317954,... (Sloane's A003010). The remainder when

is divided by

is called the Lucas-Lehmer Residue for

. The Lucas-Lehmer Residue is 0 Iff

is Prime. This test can also be extended to arbitraryIntegers.

A generalized version of the Lucas-Lehmer test lets

(2)

with

the distinct Prime factors, and

their respective Powers. If there exists a Lucas Sequence

such that

(3)

for

, ...,

and

(4)

then

is a Prime. The test is particularly simple for Mersenne Numbers, yielding the conventionalLucas-Lehmer test.

See also Lucas Sequence, Mersenne Number, Rabin-Miller Strong Pseudoprime Test

References

Sloane, N. J. A. SequenceA003010/M3494in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

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