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Profile of Michael Cameron
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Name: 
Michael Cameron 



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Date of Birth: 
8th November 1962 


Place of Birth: 
Hamilton, Ontario, Canada 


Profession: 
Actor 


From Wikipedia, the free Encyclopedia In mathematics, a Mersenne prime is a prime number that is one less than a power of two. For example, 3 = 4 − 1 = 2^{2} − 1 is a Mersenne prime; so is 7 = 8 − 1 = 2^{3} − 1. On the other hand, 15 = 16 − 1 = 2^{4} − 1, for example, is not a prime, because 15 is divisible by 3 and 5.
More generally, Mersenne numbers (not necessarily primes, but candidates for primes) are numbers that are one less than a power of two; hence,
 M_{n} = 2^{n} − 1.
Mersenne primes have a close connection to perfect numbers, which are numbers that are equal to the sum of their proper divisors. Historically, the study of Mersenne primes was motivated by this connection; in the 4th century BC Euclid demonstrated that if M is a Mersenne prime then M(M+1)/2 is a perfect number. Two millennia later, in the 18th century, Euler proved that all even perfect numbers have this form. No odd perfect numbers are known, and it is suspected that none exists.
It is currently unknown whether there is an infinite number of Mersenne primes.
Properties of Mersenne numbers
Mersenne numbers share several properties:
M_{n} is a sum of binomial coefficients: .
If a is a divisor of M_{q} (q prime) then a has the following properties: : and: .
A theorem from Euler about numbers of the form 1+6k shows that M_{q} (q prime) is a prime if and only if there exists only one pair (x,y) such that: M_{q} = (2x)^{2} + 3(3y)^{2} with . More recently, Bas Jansen has studied M_{q} = x^{2} + dy^{2} for d = 0 ... 48 and has provided a new (and clearer) proof for case d = 3.
Let be a prime. 2q + 1 is also a prime if and only if 2q + 1 divides M_{q} .
Reix has recently found that prime and composite Mersenne numbers M_{q} (q prime > 3) can be written as: M_{q} = (8x)^{2} − (3qy)^{2} = (1 + Sq)^{2} − (Dq)^{2} . Obviously, if there exists only one pair (x, y), then M_{q} is prime.
Ramanujan has showed that the equation: M_{q} = 6 + x^{2} has only 3 solutions with q prime: 3, 5, and 7 (and 2 solutions with q composite).
Searching for Mersenne primes
The calculation
shows that M_{n} can be prime only if n itself is prime, which simplifies the search for Mersenne primes considerably. But the converse is not true; M_{n} may be composite even though n is prime. For example, .
Fast algorithms for finding Mersenne primes are available, and this is why the largest known prime numbers today are Mersenne primes.
The first four Mersenne primes M_{2}, M_{3}, M_{5}, M_{7} were known in antiquity. The fifth, M_{13}, was discovered anonymously before 1461; the next two (M_{17} and M_{19}) were found by Cataldi in 1588. After more than a century M_{31} was verified to be prime by Euler in 1750. 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 numbers are named after 17th century French mathematician Marin Mersenne, who provided a list of Mersenne primes with exponents up to 257; unfortunately, his list was not correct, though, as he mistakenly included M_{67} and M_{257}, and omitted M_{61}, M_{89} and M_{107}.
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 1878 and improved by Lehmer in the 1930s, now known as the LucasLehmer test. Specifically, it can be shown that M_{n} = 2^{n} − 1 is prime if and only if M_{n} evenly divides S_{n2}, where S_{0} = 4 and for k > 0, .
The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. 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 (SWAC) 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 thirtyeight 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, and M_{44497} is the first Gigantic.
As of August 2005, only 42 Mersenne primes were known; the largest known prime number (2^{25,964,951} − 1) is a Mersenne prime. Like several previous Mersenne primes, it was discovered by a distributed computing project on the Internet, known as the Great Internet Mersenne Prime Search (GIMPS).
List of Mersenne primes
The table below lists all known Mersenne primes (sequence A000668 in OEIS):
# 
n 
M_{n} 
Digits in M_{n} 
Date of discovery 
Discoverer 
1 
2 
3 
1 
ancient 
ancient 
2 
3 
7 
1 
ancient 
ancient 
3 
5 
31 
2 
ancient 
ancient 
4 
7 
127 
3 
ancient 
ancient 
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 
Powers 
12 
127 
170141183…884105727 
39 
1876 
Lucas 
13 
521 
686479766…115057151 
157 
January 30, 1952 
Robinson 
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 
19 
4,253 
190797007…350484991 
1,281 
November 3, 1961 
Hurwitz 
20 
4,423 
285542542…608580607 
1,332 
November 3, 1961 
Hurwitz 
21 
9,689 
478220278…225754111 
2,917 
May 11, 1963 
Gillies 
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 
25 
21,701 
448679166…511882751 
6,533 
October 30, 1978 
Noll & Nickel 
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 20, 1983 
Slowinski 
31 
216,091 
746093103…815528447 
65,050 
September 6, 1985 
Slowinski 
32 
756,839 
174135906…544677887 
227,832 
February 19, 1992 
Slowinski & Gage 
33 
859,433 
129498125…500142591 
258,716 
January 10, 1994 
Slowinski & Gage 
34 
1,257,787 
412245773…089366527 
378,632 
September 3, 1996 
Slowinski & Gage 
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 Findley 
42^{*} 
25,964,951 
122164630…577077247 
7,816,230 
February 18, 2005 
GIMPS / Martin Nowak 
^{*}It is not known whether any undiscovered Mersenne primes exist between the 38th (M_{6972593}) and the 42nd (M_{25964951}) on this chart; the ranking is therefore provisional.
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