“Mersenne Prime”的概念、定义、习题解题方法及翻译-数学辞典

A Mersenne Number which is Prime is called a Mersenne prime. In order for the Mersenne number

defined by

is a Prime, then

Divides

Iff

is Prime.It is also true that Prime divisors of

must have the form

where

is a Positive Integerand simultaneously of either the form

(Uspensky and Heaslet). A Prime factor

of a Mersenne number

is a Wieferich Prime Iff

, Therefore, Mersenne Primes arenot Wieferich Primes. All known Mersenne numbers

with

Prime are Squarefree. However, Guy (1994) believes that there are

which are not Squarefree.

Trial Division is often used to establish the Compositeness of a potential Mersenne prime. This test immediately shows

to be Composite for

, 23, 83, 131, 179, 191, 239, and 251 (with small factors 23,47, 167, 263, 359, 383, 479, and 503, respectively). A much more powerful primality test for

is the Lucas-LehmerTest.

It has been conjectured that there exist an infinite number of Mersenne primes, although finding them is computationallyvery challenging. The table below gives the index

of known Mersenne primes (Sloane's A000043)

, together with thenumber of digits, discovery years, and discoverer. A similar table has been compiled by C. Caldwell. Note that the regionafter the 35th known Mersenne prime has not been completely searched, so identification of ``the'' 36thMersenne prime is tentative. L. Welsh maintains an extensive bibliography and history of Mersenne numbers. G. Woltmanhas organized a distributed search program via the Internet in which hundreds of volunteers use their personal computers toperform pieces of the search.

#		Digits	Year	Published Reference
1	2	1	Antiquity
2	3	1	Antiquity
3	5	2	Antiquity
4	7	3	Antiquity
5	13	4	1461	Reguis 1536, Cataldi 1603
6	17	6	1588	Cataldi 1603
7	19	6	1588	Cataldi 1603
8	31	10	1750	Euler 1772
9	61	19	1883	Pervouchine 1883, Seelhoff 1886
10	89	27	1911	Powers 1911
11	107	33	1913	Powers 1914
12	127	39	1876	Lucas 1876
13	521	157	1952	Lehmer 1952-3, Robinson 1952
14	607	183	1952	Lehmer 1952-3, Robinson 1952
15	1279	386	1952	Lehmer 1952-3, Robinson 1952
16	2203	664	1952	Lehmer 1952-3, Robinson 1952
17	2281	687	1952	Lehmer 1952-3, Robinson 1952
18	3217	969	1957	Riesel 1957
19	4253	1281	1961	Hurwitz 1961
20	4423	1332	1961	Hurwitz 1961
21	9689	2917	1963	Gillies 1964
22	9941	2993	1963	Gillies 1964
23	11213	3376	1963	Gillies 1964
24	19937	6002	1971	Tuckerman 1971
25	21701	6533	1978	Noll and Nickel 1980
26	23209	6987	1979	Noll 1980
27	44497	13395	1979	Nelson and Slowinski 1979
28	86243	25962	1982	Slowinski 1982
29	110503	33265	1988	Colquitt and Welsh 1991
30	132049	39751	1983	Slowinski 1988
31	216091	65050	1985	Slowinski 1989
32	756839	227832	1992	Gage and Slowinski 1992
33	859433	258716	1994	Gage and Slowinski 1994
34	1257787	378632	1996	Slowinski and Gage
35	1398269	420921	1996	Armengaud, Woltman, et al.
36?	2976221	895832	1997	Spence
37?	3021377	909526	1998	Clarkson, Woltman, et al.

Bateman, P. T.; Selfridge, J. L.; and Wagstaff, S. S. ``The New Mersenne Conjecture.'' Amer. Math. Monthly 96, 125-128, 1989.

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 66, 1987.

Beiler, A. H. Ch. 3 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966.

Caldwell, C. ``Mersenne Primes: History, Theorems and Lists.'' http://www.utm.edu/research/primes/mersenne.shtml.

Caldwell, C. ``GIMPS Finds a Prime! is Prime.'' http://www.utm.edu/research/primes/notes/1398269/.

Colquitt, W. N. and Welsh, L. Jr. ``A New Mersenne Prime.'' Math. Comput. 56, 867-870, 1991.

Conway, J. H. and Guy, R. K. ``Mersenne's Numbers.'' In The Book of Numbers. New York: Springer-Verlag, pp. 135-137, 1996.

Gillies, D. B. ``Three New Mersenne Primes and a Statistical Theory.'' Math Comput. 18, 93-97, 1964.

Guy, R. K. ``Mersenne Primes. Repunits. Fermat Numbers. Primes of Shape [sic].'' §A3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 8-13, 1994.

Haghighi, M. ``Computation of Mersenne Primes Using a Cray X-MP.'' Intl. J. Comput. Math. 41, 251-259, 1992.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 14-16, 1979.

Kraitchik, M. ``Mersenne Numbers and Perfect Numbers.'' §3.5 in Mathematical Recreations. New York: W. W. Norton, pp. 70-73, 1942.

Kravitz, S. and Berg, M. ``Lucas' Test for Mersenne Numbers .'' Math. Comput. 18, 148-149, 1964.

Lehmer, D. H. ``On Lucas's Test for the Primality of Mersenne's Numbers.'' J. London Math. Soc. 10, 162-165, 1935.

Mersenne Organization. ``GIMPS Discovers 36th Known Mersenne Prime, is Now the Largest Known Prime.'' http://www.mersenne.org/2976221.htm.

Mersenne Organization. ``GIMPS Discovers 37th Known Mersenne Prime, is Now the Largest Known Prime.'' http://www.mersenne.org/3021377.htm.

Noll, C. and Nickel, L. ``The 25th and 26th Mersenne Primes.'' Math. Comput. 35, 1387-1390, 1980.

Powers, R. E. ``The Tenth Perfect Number.'' Amer. Math. Monthly 18, 195-196, 1911.

Powers, R. E. ``Note on a Mersenne Number.'' Bull. Amer. Math. Soc. 40, 883, 1934.

Sloane, N. J. A. SequenceA000043/M0672in ``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.

Slowinski, D. ``Searching for the 27th Mersenne Prime.'' J. Recreat. Math. 11, 258-261, 1978-1979.

Tuckerman, B. ``The 24th Mersenne Prime.'' Proc. Nat. Acad. Sci. USA 68, 2319-2320, 1971.

Uhler, H. S. ``A Brief History of the Investigations on Mersenne Numbers and the Latest Immense Primes.'' Scripta Math. 18, 122-131, 1952.

Uspensky, J. V. and Heaslet, M. A. Elementary Number Theory. New York: McGraw-Hill, 1939.

Welsh, L. ``Mersenne Numbers & Mersenne Primes Bibliography.'' http://www.scruznet.com/~luke/biblio.htm.

Woltman, G. ``The GREAT Internet Mersenne Prime Search.'' http://www.mersenne.org/prime.htm.