In
mathematics, a
Mersenne
number is a positive integer that is one less than a
power of two:
- M_p=2^p-1.\,
Some definitions of Mersenne numbers require that the exponent
p be prime.
A
Mersenne prime is a Mersenne number that is
prime. It is known that if
2
^{p} − 1 is prime then
p is
prime so it makes no difference which Mersenne number definition is
used. , only 47 Mersenne primes are known; the
largest known prime number
(2
^{43,112,609} − 1) is a Mersenne prime. In modern times,
the largest known prime has almost always been a Mersenne prime.
Like several previously-discovered Mersenne primes, it was
discovered by a
distributed
computing project on the Internet, known as the
Great Internet Mersenne
Prime Search (GIMPS). It was the first known prime number
with more than 10 million base-10 digits.
About Mersenne primes
Many fundamental questions about Mersenne primes remain unresolved.
It is not even known whether the set of Mersenne primes is finite.
The
Lenstra–Pomerance–Wagstaff
conjecture asserts that, on the contrary, there are infinitely
many Mersenne primes and predicts their
order of growth. It is also not known
whether infinitely many Mersenne numbers with prime exponents are
composite, although this would
follow from widely believed conjectures about prime numbers, for
example, the infinitude of
Sophie
Germain primes.
A basic theorem about Mersenne numbers states that in order for
M_{p} to be a Mersenne prime, the
exponent
p itself must be a prime number. This rules out
primality for numbers such as
M_{4} =
2
^{4} − 1 = 15: since the exponent 4 = 2×2 is
composite, the theorem predicts that 15 is
also composite; indeed, 15 = 3×5. The three smallest Mersenne
primes are
- M_{2} = 3, M_{3} = 7,
M_{5} = 31.
While it is true that only Mersenne numbers
M_{p}, where
p = 2, 3, 5, …
could be prime, it may nevertheless turn out that
M_{p} is not prime even for a prime
exponent
p. The smallest counterexample is the Mersenne
number
- M_{11} = 2^{11} − 1 = 2047 = 23 ×
89,
which is not prime, even though 11 is a prime number. The lack of
an obvious rule to determine whether a given Mersenne number is
prime makes the search for Mersenne primes an interesting task,
which becomes difficult very quickly, since Mersenne numbers grow
very rapidly. The
Lucas–Lehmer primality
test is an efficient
primality
test that greatly aids this task. The search for the largest
known prime has somewhat of a cult following. Consequently, a lot
of computer power has been expended searching for new Mersenne
primes, much of which is now done using
distributed computing.
Mersenne primes are used in
pseudorandom number generators
such as the
Mersenne twister,
Park–Miller
random number generator, Generalized Shift Register and
Fibonacci RNG.
Searching for Mersenne primes
The identity
- 2^{ab}-1=(2^a-1)\cdot
\left(1+2^a+2^{2a}+2^{3a}+\cdots+2^{(b-1)a}\right)
shows that
M_{p} can be prime only if
p
itself is prime—that is, the primality of
p is necessary
but not sufficient for
M_{p} to be prime—which
simplifies the search for Mersenne primes considerably. The
converse statement, namely that
M_{p} is
necessarily prime if
p is prime, is false. The smallest
counterexample is 2
^{11} − 1 = 2,047 = 23×89, a
composite number.
Fast algorithms for finding Mersenne primes are available, and the
largest known prime numbers as of 2008 are Mersenne primes.
The first four Mersenne primes
M_{2} = 3,
M_{3} = 7,
M_{5} = 31 and
M
_{7} = 127 were known in antiquity. The fifth,
M_{13} = 8191, was discovered
anonymously before 1461; the next two (
M_{17} and
M_{19}) were found by
Cataldi in 1588. After nearly two centuries,
M_{31} was verified to be prime by
Euler in 1772. The next (in historical, not
numerical order) was
M_{127}, found by
Lucas in 1876, then
M_{61} by
Pervushin in 1883. Two
more (
M_{89} and
M_{107}) were
found early in the 20th century, by
Powers in 1911 and 1914, respectively.
The best method presently known for testing the primality of
Mersenne numbers is based on the computation of a recurring
sequence, as developed originally by
Lucas in 1856 and improved by
Lehmer in the 1930s, now known as the
Lucas–Lehmer primality test. Specifically, it can be shown that
(for
p > 2)
M_{p} =
2
^{p} − 1 is prime if and only if
M_{p} divides
S_{p}_{−2},
where
S_{0} = 4 and, for k > 0, S_k =
S_{k-1}^2-2.
Graph of number of digits in largest
known Mersenne prime by year - electronic era.
Note that the vertical scale is logarithmic.
The search for Mersenne primes was revolutionized by the
introduction of the electronic digital computer.
Alan Turing searched for them on the
Manchester Mark 1 in 1949. But the first
successful identification of a Mersenne prime,
M_{521}, by this means was achieved at 10:00 P.M.
on January 30, 1952 using the U.S.
National Bureau of Standards
Western Automatic Computer at the
Institute for Numerical
Analysis at the University of
California, Los Angeles, under the direction of Lehmer, with a computer search program
written and run by Prof. R.M. Robinson. It was the first Mersenne
prime to be identified in thirty-eight years; the next one,
M_{607}, was found by the computer a little less
than two hours later. Three more —
M_{1279},
M_{2203},
M_{2281} — were
found by the same program in the next several months.
M_{4253} is the first Mersenne prime that is
titanic,
M_{44497} is
the first
gigantic, and
M_{6,972,593} was the first
megaprime to be discovered, being a prime with at
least 1,000,000 digits. All three were the first known prime of any
kind of that size.
In
September 2008, mathematicians at UCLA participating in GIMPS won part of a $100,000 prize
from the Electronic
Frontier Foundation for their discovery of a very nearly
13-million-digit Mersenne prime. The prize, finally
confirmed in October 2009, is for the first known prime with at
least 10 million digits. The prime was found on a
Dell OptiPlex 745 on August 23, 2008. This is
the eighth Mersenne prime discovered at UCLA.
On April 12, 2009, a GIMPS server log reported that a 47th Mersenne
prime had possibly been found. This report was apparently
overlooked until June 4, 2009. The find was verified on June 12,
2009. The prime is 2
^{42,643,801} − 1. Although it is
chronologically the 47th Mersenne prime to be discovered, it is
less than the largest known which was the 45th to be
discovered.
Theorems about Mersenne numbers
- If 2^{p} − 1 is prime, then
p is prime.
- Proof: suppose that p is composite,
hence can be written p=a\cdot b with a and b >
1. As stated above, 2^{ab}-1=(2^a-1)\cdot
\left(1+2^a+2^{2a}+2^{3a}+\cdots+2^{(b-1)a}\right). (To check this
formula, just compute the right-hand product: most terms will
cancel out.) We have thus written 2^{ab}-1 as a product of integers
> 1, Q.E.D.
- If p is an odd prime, then any prime q that
divides 2^{p} − 1 must be 1 plus a multiple of
2p. This holds even when 2^{p} − 1 is
prime.
- Examples: Example I: 2^{5} − 1 = 31 is
prime, and 31 is 1 plus a multiple of 2×5. Example II:
2^{11} − 1 = 23×89, where 23 = 1 + 2×11, and 89 = 1 +
8×11.
- Proof: If q divides
2^{p} − 1 then 2^{p} ≡ 1 (mod
q). By Fermat's Little
Theorem, 2^{(q − 1)} ≡ 1 (mod q).
Assume p and q − 1 are relatively prime, a
similar application of Fermat's Little Theorem says that
(q − 1)^{(p − 1)} ≡ 1 (mod p).
Thus there is a number x ≡ (q −
1)^{(p − 2)} for which (q − 1)·x
≡ 1 (mod p), and therefore a number k for which
(q − 1)·x − 1 = kp. Since
2^{(q − 1)} ≡ 1 (mod q), raising both
sides of the congruence to the power
x gives 2^{(q − 1)x} ≡ 1, and
since 2^{p} ≡ 1 (mod q), raising both
sides of the congruence to the power k gives
2^{kp} ≡ 1. Thus 2^{(q −
1)x}/2^{kp} = 2^{(q −
1)x − kp} ≡ 1 (mod q). But by
definition, (q − 1)x − kp = 1, implying
that 2^{1} ≡ 1 (mod q); in other words, that
q divides 1. Thus the initial assumption that p
and q − 1 are relatively prime is untenable. Since
p is prime q − 1 must be a multiple of
p.
- If p is an odd prime, then any prime q that
divides 2^p-1 must be congruent to \pm 1 \pmod 8.
- Proof: 2^{p+1} = 2 \pmod q, so 2^{(p+1)/2} is
a square root of 2 modulo q. By quadratic reciprocity, any prime
modulo which 2 has a square root is congruent to \pm 1 \pmod
8.
- A Mersenne prime cannot be a Wieferich prime.
- Proof: We show if p=2^m-1 is a Mersenne prime,
then the congruence 2^{p-1} \equiv 1 \pmod {p^2} does not satisfy.
By Fermat's Little theorem, m |p-1. Now write, p-1=m\lambda. If the
given congruence satisfies, then p^2|2^{m\lambda}-1,therefore 0
\equiv (2^{m\lambda}-1)/(2^m-1)=1+2^{m}+2^{2m}+...+2^{(\lambda-1)m}
\equiv -\lambda \pmod {2^m-1}. Hence 2^m-1|\lambda,and therefore
\lambda \geq 2^m-1. This leads to p-1 \geq m(2^m-1), which is
impossible since m \geq 2.
History
Mersenne primes were considered already by Euclid, who found a
connection with the
perfect numbers.
They are
named after 17th century French scholar
Marin Mersenne, who compiled a list
of Mersenne primes with exponents up to 257. His list was
only partially correct, as Mersenne mistakenly included
M_{67} and
M_{257} (which are
composite), and omitted
M_{61},
M_{89}, and
M_{107} (which are
prime). Mersenne gave little indication how he came up with his
list, and its rigorous verification was completed more than two
centuries later.
List of known Mersenne primes
The table below lists all known Mersenne primes :
# |
p |
M_{p} |
Digits in M_{p} |
Date of discovery |
Discoverer |
1 |
2 |
3 |
1 |
5th century BC |
Ancient Greek mathematicians |
2 |
3 |
7 |
1 |
5th century BC |
Ancient Greek mathematicians |
3 |
5 |
31 |
2 |
3rd century BC |
Ancient Greek mathematicians |
4 |
7 |
127 |
3 |
3rd century BC |
Ancient Greek mathematicians |
5 |
13 |
8191 |
4 |
1456 |
anonymous |
6 |
17 |
131071 |
6 |
1588 |
Cataldi |
7 |
19 |
524287 |
6 |
1588 |
Cataldi |
8 |
31 |
2147483647 |
10 |
1772 |
Euler |
9 |
61 |
2305843009213693951 |
19 |
1883 |
Pervushin |
10 |
89 |
618970019…449562111 |
27 |
1911 |
Powers |
11 |
107 |
162259276…010288127 |
33 |
1914 |
PowersThe Prime Pages, M_{107}: Fauquembergue or Powers?. |
12 |
127 |
170141183…884105727 |
39 |
1876 |
Lucas |
13 |
521 |
686479766…115057151 |
157 |
January 30, 1952 |
Robinson, using SWAC |
14 |
607 |
531137992…031728127 |
183 |
January 30, 1952 |
Robinson |
15 |
1,279 |
104079321…168729087 |
386 |
June 25, 1952 |
Robinson |
16 |
2,203 |
147597991…697771007 |
664 |
October 7, 1952 |
Robinson |
17 |
2,281 |
446087557…132836351 |
687 |
October 9, 1952 |
Robinson |
18 |
3,217 |
259117086…909315071 |
969 |
September 8, 1957 |
Riesel, using BESK |
19 |
4,253 |
190797007…350484991 |
1,281 |
November 3, 1961 |
Hurwitz, using IBM 7090 |
20 |
4,423 |
285542542…608580607 |
1,332 |
November 3, 1961 |
Hurwitz |
21 |
9,689 |
478220278…225754111 |
2,917 |
May 11, 1963 |
Gillies, using ILLIAC II |
22 |
9,941 |
346088282…789463551 |
2,993 |
May 16, 1963 |
Gillies |
23 |
11,213 |
281411201…696392191 |
3,376 |
June 2, 1963 |
Gillies |
24 |
19,937 |
431542479…968041471 |
6,002 |
March 4, 1971 |
Tuckerman, using IBM 360/91 |
25 |
21,701 |
448679166…511882751 |
6,533 |
October 30, 1978 |
Noll & Nickel, using CDC
Cyber 174 |
26 |
23,209 |
402874115…779264511 |
6,987 |
February 9, 1979 |
Noll |
27 |
44,497 |
854509824…011228671 |
13,395 |
April 8, 1979 |
Nelson & Slowinski |
28 |
86,243 |
536927995…433438207 |
25,962 |
September 25, 1982 |
Slowinski |
29 |
110,503 |
521928313…465515007 |
33,265 |
January 28, 1988 |
Colquitt & Welsh |
30 |
132,049 |
512740276…730061311 |
39,751 |
September 19, 1983 |
Slowinski |
31 |
216,091 |
746093103…815528447 |
65,050 |
September 1, 1985 |
Slowinski |
32 |
756,839 |
174135906…544677887 |
227,832 |
February 19, 1992 |
Slowinski & Gage on Harwell Lab Cray-2 |
33 |
859,433 |
129498125…500142591 |
258,716 |
January 4, 1994 |
Slowinski & Gage |
34 |
1,257,787 |
412245773…089366527 |
378,632 |
September 3, 1996 |
Slowinski & GageThe Prime Pages, A Prime
of Record Size! 2^{1257787}-1. |
35 |
1,398,269 |
814717564…451315711 |
420,921 |
November 13, 1996 |
GIMPS /
Joel Armengaud |
36 |
2,976,221 |
623340076…729201151 |
895,932 |
August 24, 1997 |
GIMPS / Gordon Spence |
37 |
3,021,377 |
127411683…024694271 |
909,526 |
January 27, 1998 |
GIMPS / Roland Clarkson |
38 |
6,972,593 |
437075744…924193791 |
2,098,960 |
June 1, 1999 |
GIMPS / Nayan Hajratwala |
39 |
13,466,917 |
924947738…256259071 |
4,053,946 |
November 14, 2001 |
GIMPS / Michael Cameron |
40 |
20,996,011 |
125976895…855682047 |
6,320,430 |
November 17, 2003 |
GIMPS / Michael Shafer |
41 |
24,036,583 |
299410429…733969407 |
7,235,733 |
May 15, 2004 |
GIMPS / Josh FindleyGIMPS, Mersenne.org
Project Discovers New Largest Known Prime Number,
2^{24,036,583}-1. |
42 |
25,964,951 |
122164630…577077247 |
7,816,230 |
February 18, 2005 |
GIMPS / Martin NowakGIMPS, Mersenne.org
Project Discovers New Largest Known Prime Number,
2^{25,964,951}-1. |
43 |
30,402,457 |
315416475…652943871 |
9,152,052 |
December 15, 2005 |
GIMPS / Curtis
Cooper & Steven BooneGIMPS,
Mersenne.org Project Discovers New Largest Known Prime
Number, 2^{30,402,457}-1. |
44 |
32,582,657 |
124575026…053967871 |
9,808,358 |
September 4, 2006 |
GIMPS / Curtis Cooper & Steven BooneGIMPS, Mersenne.org
Project Discovers Largest Known Prime Number,
2^{32,582,657}-1. |
45 |
37,156,667 |
202254406…308220927 |
11,185,272 |
September 6, 2008 |
GIMPS / Hans-Michael Elvenich |
46 |
42,643,801 |
169873516…562314751 |
12,837,064 |
April 12, 2009 |
GIMPS / Odd M. Strindmo |
47 |
43,112,609 |
316470269…697152511 |
12,978,189 |
August 23, 2008 |
GIMPS / Edson Smith |
It is not known whether any undiscovered Mersenne primes exist between the 39th (M_{13,466,917}) and the 47th (M_{43,112,609}) on this chart; the ranking is therefore provisional. Primes are not always discovered in increasing order. For example, the 29th Mersenne prime was discovered after the 30th and the 31st. Similarly, the current record holder was followed by 2 smaller Mersenne primes, first 2 weeks later and then 8 months later.
M_{42,643,801} was first found by a machine on April 12, 2009; however, no human took notice of this fact until June 4. Thus, either April 12 or June 4 may be considered the 'discovery' date. The discoverer, Strindmo, apparently used the alias Stig M. Valstad.
To help visualize the size of the 47th known Mersenne prime, it
would require 3,461 pages to display the number in base 10 with 75
digits per line and 50 lines per page.
Factorization of Mersenne numbers
The factorization of a prime number is by definition the number
itself. This section is about composite numbers. Mersenne numbers
are very good test cases for the
special number field sieve
algorithm, so often the largest number factorized with this
algorithm has been a Mersenne number.
, 2^{1039} − 1
is the record-holder, after a calculation taking about a year on a
couple of hundred computers, mostly at NTT in
Japan and at EPFL in
Switzerland. See
integer factorization records
for links to more information. The special number field sieve can
factorize numbers with more than one large factor. If a number has
only one very large factor then other algorithms can factorize
larger numbers by first finding small factors and then making a
primality test on the cofactor. , the
largest composite Mersenne number with proven prime factors is
2
^{17029} − 1 = 418879343 ×
p, where
p
was proven prime with
ECPP. The largest with
probable prime factors allowed is
2
^{684127} − 1 = 23765203727 ×
q, where
q
is a probable prime.
Perfect numbers
Mersenne primes are interesting to many for their connection to
perfect numbers. In the 4th century
BC,
Euclid demonstrated that if
M_{p} is a Mersenne prime then
- 2^{p−1}×(2^{p} − 1) =
M_{p}(M_{p} +
1)/2
is an
even perfect number (which is also
the
M_{p}th
triangular number). In the 18th century,
Leonhard Euler proved that,
conversely, all even perfect numbers have this form. It is unknown
whether there are any odd perfect numbers, but it appears unlikely
that there is one.
Generalization
The
binary representation of
2
^{p} − 1 is the digit 1 repeated
p
times, for example, 2
^{5} − 1 = 11111
_{2} in the
binary notation. A Mersenne number is therefore a
repunit in base 2, and Mersenne primes are the base
2 repunit primes.
The base 2 representation of a Mersenne number shows the
factorization pattern for composite exponent. For example:
- M_{35}=2^{35}-1=(11111111111111111111111111111111111)_2\,
- =(11111)_2 \cdot (1000010000100001000010000100001)_2=M_5 \cdot
(1000010000100001000010000100001)_2\,
- =(1111111)_2 \cdot (10000001000000100000010000001)_2=M_7 \cdot
(10000001000000100000010000001)_2\,
- =(11111)_2 \cdot (1111111)_2 \cdot
[(1000010100101010010100001)_2 -
(0100001010010100101000010)_2]\,
- =M_5 \cdot M_7 \cdot (100001010010101101011111)_2\,
Mersenne numbers in nature and elsewhere
In computer science, unsigned
p-bit
integers can be used to express
numbers up to
M_{p}.
In the mathematical problem
Tower of
Hanoi, solving a puzzle with a
p-disc tower requires
at least
M_{p} steps.
See also
References
- 12-million-digit prime number sets record, nets
$100,000 prize
- The largest known prime has been a Mersenne prime since 1952,
except between 1989 and 1992; see Caldwell, " The
Largest Known Prime by Year: A Brief History" from the
Prime Pages
website, University of Tennessee at
Martin.
- The Prime Pages, The
Largest Known Prime by Year: A Brief History.
- Prime Curios!, 17014...05727 (39-digits).
- Brian Napper, The
Mathematics Department and the Mark 1.
- The Prime Pages, The Prime Glossary: megaprime.
- UCLA mathematicians discover a 13-million-digit
prime number, Los Angeles Times, September 27, 2008
- The Prime Pages, Mersenne's conjecture.
- The Prime Pages, Mersenne Primes: History, Theorems and
Lists.
- Landon
Curt Noll, Mersenne Prime Digits and Names.
- The Prime Pages, The
finding of the 32nd Mersenne.
- Chris Caldwell, The Largest Known Primes.
- GIMPS Discovers 35th Mersenne Prime.
- GIMPS Discovers 36th Known Mersenne Prime.
- GIMPS Discovers 37th Known Mersenne Prime.
- GIMPS Finds First Million-Digit Prime, Stakes Claim to
$50,000 EFF Award.
- GIMPS, Researchers Discover Largest Multi-Million-Digit Prime
Using Entropia Distributed Computing Grid.
- GIMPS, Mersenne Project Discovers Largest Known Prime Number on
World-Wide Volunteer Computer Grid.
- Titanic Primes Raced to Win $100,000 Research Award.
Retrieved on 2008-09-16.
- Paul
Zimmermann, "Integer Factoring Records".
- Chris Caldwell, The
Top Twenty: Mersenne cofactor at The Prime Pages.
- Donovan Johnson, "Largest
known probable prime Mersenne Cofactors".
External links
MathWorld links