== M_23,209 is 36 CPU hours ≡ 1979 == Welcome to 1979! :Yeah. I know. :-) --[[User:Short Circuit|Short Circuit]] 07:55, 21 February 2008 (MST)

==List of known Mersenne primes== The table below lists all known Mersenne primes: {| class="wikitable" |- ! # ! ''n'' ! ''M''_''n'' ! Digits in ''M''_''n'' ! Date of discovery ! Discoverer |- | align="right" | 1 | align="right" | 2 | align="right" | 3 (number)|3 | align="right" | 1 | ''ancient'' | ''ancient'' |- | align="right" | 2 | align="right" | 3 | align="right" | 7 (number)|7 | align="right" | 1 | ''ancient'' | ''ancient'' |- | align="right" | 3 | align="right" | 5 | align="right" | 31 (number)|31 | align="right" | 2 | ''ancient'' | ''ancient'' |- | align="right" | 4 | align="right" | 7 | align="right" | 127 (number)|127 | align="right" | 3 | ''ancient'' | ''ancient'' |- | align="right" | 5 | align="right" | 13 | align="right" | 8191 | align="right" | 4 | 1456 | ''anonymous'' [http://primes.utm.edu/mersenne/] |- | align="right" | 6 | align="right" | 17 | align="right" | 131071 | align="right" | 6 | 1588 | Pietro Cataldi|Cataldi |- | align="right" | 7 | align="right" | 19 | align="right" | 524287 | align="right" | 6 | 1588 | Pietro Cataldi|Cataldi |- | align="right" | 8 | align="right" | 31 | align="right" | 2147483647 | align="right" | 10 | 1772 | Leonhard Euler|Euler |- | align="right" | 9 | align="right" | 61 | align="right" | 2305843009213693951 | align="right" | 19 | 1883 | Ivan Mikheevich Pervushin|Pervushin |- | align="right" | 10 | align="right" | 89 | align="right" | 618970019…449562111 | align="right" | 27 | 1911 | R. E. Powers|Powers |- | align="right" | 11 | align="right" | 107 | align="right" | 162259276…010288127 | align="right" | 33 | 1914 | R. E. Powers|Powers[http://primes.utm.edu/notes/fauquem.html] |- | align="right" | 12 | align="right" | 127 | align="right" | 170141183…884105727 | align="right" | 39 | 1876 | Edouard Lucas|Lucas |- | align="right" | 13 | align="right" | 521 | align="right" | 686479766…115057151 | align="right" | 157 | January 30 1952 | Raphael M. Robinson|Robinson |- | align="right" | 14 | align="right" | 607 | align="right" | 531137992…031728127 | align="right" | 183 | January 30 1952 | Raphael M. Robinson|Robinson |- | align="right" | 15 | align="right" | 1,279 | align="right" | 104079321…168729087 | align="right" | 386 | June 25 1952 | Raphael M. Robinson|Robinson |- | align="right" | 16 | align="right" | 2,203 | align="right" | 147597991…697771007 | align="right" | 664 | October 7 1952 | Raphael M. Robinson|Robinson |- | align="right" | 17 | align="right" | 2,281 | align="right" | 446087557…132836351 | align="right" | 687 | October 9 1952 | Raphael M. Robinson|Robinson |- | align="right" | 18 | align="right" | 3,217 | align="right" | 259117086…909315071 | align="right" | 969 | September 8 1957 | Hans Riesel|Riesel |- | align="right" | 19 | align="right" | 4,253 | align="right" | 190797007…350484991 | align="right" | 1,281 | November 3 1961 | Alexander Hurwitz|Hurwitz |- | align="right" | 20 | align="right" | 4,423 | align="right" | 285542542…608580607 | align="right" | 1,332 | November 3 1961 | Alexander Hurwitz|Hurwitz |- | align="right" | 21 | align="right" | 9,689 | align="right" | 478220278…225754111 | align="right" | 2,917 | May 11 1963 | Donald B. Gillies|Gillies |- | align="right" | 22 | align="right" | 9,941 | align="right" | 346088282…789463551 | align="right" | 2,993 | May 16 1963 | Donald B. Gillies|Gillies |- | align="right" | 23 | align="right" | 11,213 | align="right" | 281411201…696392191 | align="right" | 3,376 | June 2 1963 | Donald B. Gillies|Gillies |- | align="right" | 24 | align="right" | 19,937 | align="right" | 431542479…968041471 | align="right" | 6,002 | March 4 1971 | Bryant Tuckerman|Tuckerman |- | align="right" | 25 | align="right" | 21,701 | align="right" | 448679166…511882751 | align="right" | 6,533 | October 30 1978 | Landon Curt Noll|Noll & Laura Nickel|Nickel |- | align="right" | 26 | align="right" | 23,209 | align="right" | 402874115…779264511 | align="right" | 6,987 | February 9 1979 | Landon Curt Noll|Noll |- | align="right" | 27 | align="right" | 44,497 | align="right" | 854509824…011228671 | align="right" | 13,395 | April 8 1979 | Harry Nelson|Nelson & David Slowinski|Slowinski |- | align="right" | 28 | align="right" | 86,243 | align="right" | 536927995…433438207 | align="right" | 25,962 | September 25 1982 | David Slowinski|Slowinski |- | align="right" | 29 | align="right" | 110,503 | align="right" | 521928313…465515007 | align="right" | 33,265 | January 28 1988 | Walt Colquitt|Colquitt & Luke Welsh|Welsh |- | align="right" | 30 | align="right" | 132,049 | align="right" | 512740276…730061311 | align="right" | 39,751 | September 19 1983[http://www.isthe.com/chongo/tech/math/prime/mersenne.html#largest] | David Slowinski|Slowinski |- | align="right" | 31 | align="right" | 216,091 | align="right" | 746093103…815528447 | align="right" | 65,050 | September 1 1985[http://www.isthe.com/chongo/tech/math/prime/mersenne.html#largest] | David Slowinski|Slowinski |- | align="right" | 32 | align="right" | 756,839 | align="right" | 174135906…544677887 | align="right" | 227,832 | February 19 1992 | David Slowinski|Slowinski & Paul Gage|Gage on Harwell Lab Cray-2 [http://primes.utm.edu/notes/756839.html] |- | align="right" | 33 | align="right" | 859,433 | align="right" | 129498125…500142591 | align="right" | 258,716 | January 4 1994 [http://www.math.unicaen.fr/~reyssat/largest.html] | David Slowinski|Slowinski & Paul Gage|Gage |- | align="right" | 34 | align="right" | 1,257,787 | align="right" | 412245773…089366527 | align="right" | 378,632 | September 3 1996 | David Slowinski|Slowinski & Paul Gage|Gage [http://primes.utm.edu/notes/1257787.html] |- | align="right" | 35 | align="right" | 1,398,269 | align="right" | 814717564…451315711 | align="right" | 420,921 | November 13 1996 | Great Internet Mersenne Prime Search|GIMPS / Joel Armengaud [http://www.mersenne.org/1398269.htm] |- | align="right" | 36 | align="right" | 2,976,221 | align="right" | 623340076…729201151 | align="right" | 895,932 | August 24 1997 | Great Internet Mersenne Prime Search|GIMPS / Gordon Spence [http://www.mersenne.org/2976221.htm] |- | align="right" | 37 | align="right" | 3,021,377 | align="right" | 127411683…024694271 | align="right" | 909,526 | January 27 1998 | Great Internet Mersenne Prime Search|GIMPS / Roland Clarkson [http://www.mersenne.org/3021377.htm] |- | align="right" | 38 | align="right" | 6,972,593 | align="right" | 437075744…924193791 | align="right" | 2,098,960 | June 1 1999 | Great Internet Mersenne Prime Search|GIMPS / Nayan Hajratwala [http://www.mersenne.org/6972593.htm] |- | align="right" | 39 | align="right" | 13,466,917 | align="right" | 924947738…256259071 | align="right" | 4,053,946 | November 14 2001 | Great Internet Mersenne Prime Search|GIMPS / Michael Cameron [http://www.mersenne.org/13466917.htm] |- | align="right" | 40 | align="right" | 20,996,011 | align="right" | 125976895…855682047 | align="right" | 6,320,430 | November 17 2003 | Great Internet Mersenne Prime Search|GIMPS / Michael Shafer [http://www.mersenne.org/20996011.htm] |- | align="right" | 41 | align="right" | 24,036,583 | align="right" | 299410429…733969407 | align="right" | 7,235,733 | May 15 2004 | Great Internet Mersenne Prime Search|GIMPS / Josh Findley [http://www.mersenne.org/24036583.htm] |- | align="right" | 42 | align="right" | 25,964,951 | align="right" | 122164630…577077247 | align="right" | 7,816,230 | February 18 2005 | Great Internet Mersenne Prime Search|GIMPS / Martin Nowak [http://www.mersenne.org/25964951.htm] |- | align="right" | 43 | align="right" | 30,402,457 | align="right" | 315416475…652943871 | align="right" | 9,152,052 | December 15 2005 | Great Internet Mersenne Prime Search|GIMPS / Curtis Cooper (mathematician)|Curtis Cooper & Steven Boone [http://www.mersenne.org/30402457.htm] |- | align="right" | 44^* | align="right" | 32,582,657 | align="right" | 124575026…053967871 | align="right" | 9,808,358 | September 4 2006 | Great Internet Mersenne Prime Search|GIMPS / Curtis Cooper (mathematician)|Curtis Cooper & Steven Boone [http://www.mersenne.org/32582657.htm] |}

:The 45th and 46th Mersenne primes have been discovered, is the intention to keep this table up to date? --[[User:Lupus|Lupus]] 17:24, 2 December 2008 (UTC)

:: The above list seems to be copied ''in toto'' from the updated Wikipedia site: https://en.wikipedia.org/wiki/Mersenne_prime#List_of_known_Mersenne_primes :: The above Wikipedia site now has a list of '''51''' Mersenne primes. -- [[User:Gerard Schildberger|Gerard Schildberger]] ([[User talk:Gerard Schildberger|talk]]) 19:57, 7 May 2019 (UTC)

== Java precision == The Java version is still limited by types. Integer.parseInt(args[0]) limits p to 2147483647. Also the fact that isMersennePrime takes an int limits it there too. For full arbitrary precision every int needs to be a BigInteger or BigDecimal and a square root method will need to be created for them. The limitation is OK I think (I don't think we'll be getting up to 2^2147483647 - 1 anytime soon), but the claim "any arbitrary prime" is false because of the use of ints. --[[User:Mwn3d|Mwn3d]] 07:45, 21 February 2008 (MST)

== Speeding things up == The main loop in Lucas-Lehmer is doing (n*n) mod M where M=2^p-1, and p > 1. '''But we can do it without division'''.

We compute the remainder of a division by M. Now, intuitively, dividing by 2^p-1 is almost like dividing by 2^p, except the latter is much faster since it's a shift. Let's compute how much the two divisions differ.

We will call S = nn. Notice that since the remainder mod M is computed again and again, the value of n must be < M at the beginning of a loop, that is at most 2^p-2, thus S = nn <= 2^(2p) - 42^p + 4 = 2^p * (2^p - 2) + 4 - 2*2^p

When dividing S by M, you get quotient q1 and remainder r1 with S = q1M + r1 and 0 <= r1 < M When dividing S by M+1, you get likewise S = q2(M+1) + r2 and 0 <= r2 <= M In the latter, we divide by a larger number, so the quotient must be less, or maybe equal, that is, q2 <= q1.

Subtract the two equalities, giving 0 = (q2 - q1)*M + q2 + r2 - r1 (q1 - q2)*M = q2 + r2 - r1

Since S = 2^p * (2^p - 2) + 4 - 2*2^p <= 2^p * (2^p - 2), then the quotient q2 is less than 2^p - 2 (remember, when computing q2, we divide by M+1 = 2^p).

Now, 0 <= q2 <= 2^p - 2 0 <= r2 <= 2^p - 1 0 <= r1 <= 2^p - 2 Thus the right hand side is >= 0, and <= 2*2^p - 3. The left hand side is a multiple of M = 2^p - 1.

Therefore, this multiple must be 0M or 1M, certainly not 2M = 22^p - 2, which would be > 2*2^p - 3, and not any other higher multiple would do. So we have proved that q1 - q2 = 0 or 1.

This means that division by 2^p is almost equivalent (regarding the quotient) to dividing by 2^p-1: it's the same quotient, or maybe too short by 1.

Now, the remainder S mod M. We start with a quotient q = S div 2^p, or simply q = S >> p (right shift). The remainder is S - qM = S - q(2^p - 1) = S - q*2^p + q, and the multiplication by 2^p is a left shift. And this remainder may be a bit too large, if our quotient is a bit too small (by one): in this case we must subtract M.

So, in pseudo-code, we are done if we do:

S = n*n
q = S >> p
r = S - (q << p) + q
if r >= M then r = r - M

We can go a bit further: taking S >> p then q << p is simply keeping the higher bits of S. But then we subtract these higher bits from S, so we only keep the lower bits, that is we do (S & mask), and this mask is simply M ! (remember, M = 2^p - 1, a bit mask of p bits equal to "1")

The pseudo-code can thus be written

S = n*n
r = (S & M) + (S >> p)
if r >= M then r = r - M

And we have computed a remainder mod M without any division, only a few addition/subtraction/shift/bitwise-and, which will be much faster (each has a linear time complexity).

How much faster ? For exponents between 1 and 2000, in Python, the job is done 2.87 times as fast. For exponents between 1 and 5000, it's 3.42 times as fast. And it gets better and better, since the comlexity is lower.

[[User:Arbautjc|Arbautjc]] ([[User talk:Arbautjc|talk]]) 22:04, 15 November 2013 (UTC)


May 2015 - fixed small bugs in both Python implementations. In the first, execution failed (Python 3) without a cast to int in the test. In the second, there was a typo - an 'r' should have been 's'.

: Timing for some solutions for 2..11213:
{| class="wikitable"
|-
! Time (s)
! Solution
|-
| 871.2
| Python without optimizations
|-
| 314.7
| Python with optimizations
|-
| 124.7
| Perl Math::GMP without optimizations
|-
| 106.9
| Pari/GP 2.8.0
|-
| 61.8
| Perl Math::GMP with optimization
|-
| 33.0
| Python using gmpy2 (skipping non-primes)
|-
| 14.2
| C/GMP with even more optimizations
|-
| 13.3
| Perl Math::Prime::Util::GMP (source of C/GMP code)
|}
[[User:Danaj|Danaj]] ([[User talk:Danaj|talk]]) 19:49, 9 May 2015 (UTC)

Rust solution for 2..11213 -> 8.6 s