# The Mersenne Primes

## by Stu Savory, 22/2/2005.

A Mersenne prime is a Mersenne number, i.e., a number of the form M=2n-1 which is prime.
In order for M=2n-1 to be prime, n must itself be prime, and not e.g. 15 (=3*5) or 63 (=3*3*7) which are composite.
For composite n with factors r and s, we have n = rs.
Therefore, M=2n-1 is M=2rs-1 , which is a binomial which always has factors (2s-1)*(2r+1) and so is NOT prime (Q.E.D)

As of 22/2/2005 the 42nd Mersenne Prime has (probably) been discovered. Confirmation still needed.
The currently known Mersenne Primes are :-

```Number	Power(n)	M digits	When?	Who found it?
------------------------------------------------------------------
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, Woltman, GIMPS (Devlin 1997)
37	3021377		909526		1998	Clarkson, Woltman, Kurowski, GIMPS
38	6972593		2098960		1999	Hajratwala, Woltman, Kurowski, GIMPS
39	13466917	4053946		2001	Cameron, Woltman, GIMPS
40?	20996011	6320430		2003	Shafer, GIMPS (Weisstein 2003ab)
41?	24036583	7235733		2004	Findley, GIMPS (Weisstein 2004)
42?	?		<10000000	2005	GIMP
========================================================================
```

Now go visit my blog please, or look at other interesting maths stuff :-)

 Index/Home Impressum Sitemap Search