Talk:Lucas-Lehmer test
Contents
M23,209 is 36 CPU hours ≡ 1979[edit]
Welcome to 1979!
- Yeah. I know. :-) --Short Circuit 07:55, 21 February 2008 (MST)
List of known Mersenne primes[edit]
The table below lists all known Mersenne primes:
# | n | M_{n} | Digits in M_{n} | Date of discovery | Discoverer |
---|---|---|---|---|---|
1 | 2 | 3 (number)|3 | 1 | ancient | ancient |
2 | 3 | 7 (number)|7 | 1 | ancient | ancient |
3 | 5 | 31 (number)|31 | 2 | ancient | ancient |
4 | 7 | 127 (number)|127 | 3 | ancient | ancient |
5 | 13 | 8191 | 4 | 1456 | anonymous [1] |
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[2] |
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 & Laura Nickel|Nickel |
26 | 23,209 | 402874115…779264511 | 6,987 | February 9 1979 | Noll |
27 | 44,497 | 854509824…011228671 | 13,395 | April 8 1979 | Nelson & David Slowinski|Slowinski |
28 | 86,243 | 536927995…433438207 | 25,962 | September 25 1982 | Slowinski |
29 | 110,503 | 521928313…465515007 | 33,265 | January 28 1988 | Colquitt & Luke Welsh|Welsh |
30 | 132,049 | 512740276…730061311 | 39,751 | September 19 1983[3] | Slowinski |
31 | 216,091 | 746093103…815528447 | 65,050 | September 1 1985[4] | Slowinski |
32 | 756,839 | 174135906…544677887 | 227,832 | February 19 1992 | Slowinski & Paul Gage|Gage on Harwell Lab Cray-2 [5] |
33 | 859,433 | 129498125…500142591 | 258,716 | January 4 1994 [6] | Slowinski & Paul Gage|Gage |
34 | 1,257,787 | 412245773…089366527 | 378,632 | September 3 1996 | Slowinski & Paul Gage|Gage [7] |
35 | 1,398,269 | 814717564…451315711 | 420,921 | November 13 1996 | GIMPS / Joel Armengaud [8] |
36 | 2,976,221 | 623340076…729201151 | 895,932 | August 24 1997 | GIMPS / Gordon Spence [9] |
37 | 3,021,377 | 127411683…024694271 | 909,526 | January 27 1998 | GIMPS / Roland Clarkson [10] |
38 | 6,972,593 | 437075744…924193791 | 2,098,960 | June 1 1999 | GIMPS / Nayan Hajratwala [11] |
39 | 13,466,917 | 924947738…256259071 | 4,053,946 | November 14 2001 | GIMPS / Michael Cameron [12] |
40^{*} | 20,996,011 | 125976895…855682047 | 6,320,430 | November 17 2003 | GIMPS / Michael Shafer [13] |
41^{*} | 24,036,583 | 299410429…733969407 | 7,235,733 | May 15 2004 | GIMPS / Josh Findley [14] |
42^{*} | 25,964,951 | 122164630…577077247 | 7,816,230 | February 18 2005 | GIMPS / Martin Nowak [15] |
43^{*} | 30,402,457 | 315416475…652943871 | 9,152,052 | December 15 2005 | GIMPS / Curtis Cooper (mathematician)|Curtis Cooper & Steven Boone [16] |
44^{*} | 32,582,657 | 124575026…053967871 | 9,808,358 | September 4 2006 | GIMPS / Curtis Cooper (mathematician)|Curtis Cooper & Steven Boone [17] |
- The 45th and 46th Mersenne primes have been discovered, is the intention to keep this table up to date? --Lupus 17:24, 2 December 2008 (UTC)
Java precision[edit]
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. --Mwn3d 07:45, 21 February 2008 (MST)
Speeding things up[edit]
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 = n*n. 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 = n*n <= 2^(2*p) - 4*2^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 = q1*M + 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 0*M or 1*M, certainly not 2*M = 2*2^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 - q*M = 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.
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:
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) |
Danaj (talk) 19:49, 9 May 2015 (UTC)
Rust solution for 2..11213 -> 8.6 s