“Mersenne Number”的概念、定义、习题解题方法及翻译-数学辞典

单词

Mersenne Number

释义

Mersenne Number

A number of the form

(1)

for

an Integer is known as a Mersenne number. The Mersenne numbers are therefore 2-Repdigits, and also the numbers obtained by setting

in a Fermat Polynomial. The first few are 1, 3, 7, 15, 31, 63, 127, 255, ... (Sloane's A000225).

The number of digits in the Mersenne number is

(2)

where

is the Floor Function, which, for large

, gives

(3)

In order for the Mersenne number to be Prime, must be Prime. This is true since for Composite withfactors and , . Therefore, can be written as , which is a Binomial Number and can befactored. Since the most interest in Mersenne numbers arises from attempts to factor them, many authors prefer to define aMersenne number as a number of the above form

(4)

but with

restricted to Prime values.

The search for Mersenne Primes is one of the most computationally intensive and actively pursuedareas of advanced and distributed computing.

See also Cunningham Number, Eberhart's Conjecture, Fermat Number, Lucas-Lehmer Test, MersennePrime, Perfect Number, Repunit, Riesel Number, Sierpinski Number of the SecondKind, Sophie Germain Prime, Superperfect Number, Wieferich Prime

References

Pappas, T. ``Mersenne's Number.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 211, 1989.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 14, 18-19, 22, and 29-30, 1993.

Sloane, N. J. A. SequenceA000225/M2655in ``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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