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 ========================================================================
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