42nd Mersenne Prime (Probably) Discovered

Less than a year after the 41st Mersenne prime was reported (MathWorld headline news: June 1, 2004), Great Internet Mersenne Prime Search (GIMPS) project organizer George Woltman is reporting in a Feb. 18 email to the GIMPS mailing list that a new Mersenne number has been flagged as prime and reported to the project’s server. If verified, this would be the 42nd known Mersenne prime, as well as the largest prime number known of any kind.

Mersenne numbers are numbers of the form Mn = 2n – 1, giving the first few as 1, 3, 7, 15, 31, 63, 127, …. Interestingly, the definition of these numbers therefore means that the nth Mersenne number is simply a string of n 1s when represented in binary. For example, M7 = 27 – 1 = 127 = 11111112 is a Mersenne number. In fact, since 127 is also prime, 127 is also a Mersenne prime.

The study of such numbers has a long and interesting history,
and the search for Mersenne numbers that are prime has been
a computationally challenging exercise requiring the world’s
fastest computers. Mersenne primes are intimately connected
with so-called perfect numbers,
which were extensively studied by the ancient Greeks, including by
Euclid. A complete list of indices n of the previously known
Mersenne primes is given in the table below (as well as by sequence href=”http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000043″
target=”_blank”>A000043 in Neil Sloane’s On-Line Encyclopedia
of Integer Sequences
). The last of these has a whopping 7,235,733
decimal digits. However, note that the
region between the 39th and 40th known Mersenne primes has not
been completely searched, so it is not known if M style=”vertical-align:sub, font-size:10pt;”>20,996,011

is actually the 40th Mersenne prime.

















































































































































































































# p digits year discoverer (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, 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 (Whitehouse 2001, Weisstein 2001ab)
40? 20996011 6320430 2003 Shafer, GIMPS (Weisstein 2003ab)
41? 24036583 7235733 2004 Findley, GIMPS (Weisstein 2004)
42? ? <10000000 2005 GIMPS

The eight largest known Mersenne primes (including the latest candidate)
have all been discovered by GIMPS, which is a distributed computing
project being undertaken by an international collaboration of volunteers.
Thus far, GIMPS participants have tested and double-checked all exponents
n below 9,889,900, while all exponents below 14,135,900 have been
tested at least once. Although the candidate prime was flagged prime by
an experienced GIMPS volunteer, it has yet to be verified by independent
software running on different hardware. If confirmed, GIMPS will make
an official press release that will reveal the number and the name of
the lucky discoverer.

While the exact exponent of the new find has not yet been made public,
GIMPS organizer George Woltman reported that if the new candidate is
confirmed, it would be the largest known prime, which would mean it has
7,235,733 or more digits. Woltman also noted that it has fewer than
10 million digits (a holy grail for prime searchers), meaning that the
new candidate has exponent n somewhere between 24,036,584 and
33,219,253. Woltman is currently attempting to reproduce the find from
the user's save file, thus eliminating any chance of the report
being erroneous.

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