Talk:Lucas-Lehmer test: Difference between revisions
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==
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"
|-
Line 14 ⟶ 16:
| align="right" | 1
| align="right" | 2
| align="right" |
| align="right" | 1
| ''ancient''
Line 21 ⟶ 23:
| align="right" | 2
| align="right" | 3
| align="right" |
| align="right" | 1
| ''ancient''
Line 28 ⟶ 30:
| align="right" | 3
| align="right" | 5
| align="right" |
| align="right" | 2
| ''ancient''
Line 35 ⟶ 37:
| align="right" | 4
| align="right" | 7
| align="right" |
| align="right" | 3
| ''ancient''
Line 52 ⟶ 54:
| align="right" | 6
| 1588
|
|-
| align="right" | 7
Line 59 ⟶ 61:
| align="right" | 6
| 1588
|
|-
| align="right" | 8
Line 66 ⟶ 68:
| align="right" | 10
| 1772
|
|-
| align="right" | 9
Line 73 ⟶ 75:
| align="right" | 19
| 1883
|
|-
| align="right" | 10
Line 80 ⟶ 82:
| align="right" | 27
| 1911
|
|-
| align="right" | 11
Line 87 ⟶ 89:
| align="right" | 33
| 1914
|
|-
| align="right" | 12
Line 94 ⟶ 96:
| align="right" | 39
| 1876
|
|-
| align="right" | 13
Line 100 ⟶ 102:
| align="right" | 686479766…115057151
| align="right" | 157
|
|
|-
| align="right" | 14
Line 107 ⟶ 109:
| align="right" | 531137992…031728127
| align="right" | 183
|
|
|-
| align="right" | 15
Line 114 ⟶ 116:
| align="right" | 104079321…168729087
| align="right" | 386
|
|
|-
| align="right" | 16
Line 121 ⟶ 123:
| align="right" | 147597991…697771007
| align="right" | 664
|
|
|-
| align="right" | 17
Line 128 ⟶ 130:
| align="right" | 446087557…132836351
| align="right" | 687
|
|
|-
| align="right" | 18
Line 135 ⟶ 137:
| align="right" | 259117086…909315071
| align="right" | 969
|
|
|-
| align="right" | 19
Line 142 ⟶ 144:
| align="right" | 190797007…350484991
| align="right" | 1,281
|
|
|-
| align="right" | 20
Line 149 ⟶ 151:
| align="right" | 285542542…608580607
| align="right" | 1,332
|
|
|-
| align="right" | 21
Line 156 ⟶ 158:
| align="right" | 478220278…225754111
| align="right" | 2,917
|
|
|-
| align="right" | 22
Line 163 ⟶ 165:
| align="right" | 346088282…789463551
| align="right" | 2,993
|
|
|-
| align="right" | 23
Line 170 ⟶ 172:
| align="right" | 281411201…696392191
| align="right" | 3,376
|
|
|-
| align="right" | 24
Line 177 ⟶ 179:
| align="right" | 431542479…968041471
| align="right" | 6,002
|
|
|-
| align="right" | 25
Line 184 ⟶ 186:
| align="right" | 448679166…511882751
| align="right" | 6,533
|
|
|-
| align="right" | 26
Line 191 ⟶ 193:
| align="right" | 402874115…779264511
| align="right" | 6,987
|
|
|-
| align="right" | 27
Line 198 ⟶ 200:
| align="right" | 854509824…011228671
| align="right" | 13,395
|
|
|-
| align="right" | 28
Line 205 ⟶ 207:
| align="right" | 536927995…433438207
| align="right" | 25,962
|
|
|-
| align="right" | 29
Line 212 ⟶ 214:
| align="right" | 521928313…465515007
| align="right" | 33,265
|
|
|-
| align="right" | 30
Line 219 ⟶ 221:
| align="right" | 512740276…730061311
| align="right" | 39,751
|
|
|-
| align="right" | 31
Line 226 ⟶ 228:
| align="right" | 746093103…815528447
| align="right" | 65,050
|
|
|-
| align="right" | 32
Line 233 ⟶ 235:
| align="right" | 174135906…544677887
| align="right" | 227,832
|
|
|-
| align="right" | 33
Line 240 ⟶ 242:
| align="right" | 129498125…500142591
| align="right" | 258,716
|
|
|-
| align="right" | 34
Line 247 ⟶ 249:
| align="right" | 412245773…089366527
| align="right" | 378,632
|
|
|-
| align="right" | 35
Line 254 ⟶ 256:
| align="right" | 814717564…451315711
| align="right" | 420,921
|
|
|-
| align="right" | 36
Line 261 ⟶ 263:
| align="right" | 623340076…729201151
| align="right" | 895,932
|
|
|-
| align="right" | 37
Line 268 ⟶ 270:
| align="right" | 127411683…024694271
| align="right" | 909,526
|
|
|-
| align="right" | 38
Line 275 ⟶ 277:
| align="right" | 437075744…924193791
| align="right" | 2,098,960
|
|
|-
| align="right" | 39
Line 282 ⟶ 284:
| align="right" | 924947738…256259071
| align="right" | 4,053,946
|
|
|-
| align="right" | 40<sup>*</sup>
Line 289 ⟶ 291:
| align="right" | 125976895…855682047
| align="right" | 6,320,430
|
|
|-
| align="right" | 41<sup>*</sup>
Line 296 ⟶ 298:
| align="right" | 299410429…733969407
| align="right" | 7,235,733
|
|
|-
| align="right" | 42<sup>*</sup>
Line 303 ⟶ 305:
| align="right" | 122164630…577077247
| align="right" | 7,816,230
|
|
|-
| align="right" | 43<sup>*</sup>
Line 310 ⟶ 312:
| align="right" | 315416475…652943871
| align="right" | 9,152,052
|
|
|-
| align="right" | 44<sup>*</sup>
Line 317 ⟶ 319:
| align="right" | 124575026…053967871
| align="right" | 9,808,358
|
|
|}
: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:
<big> https://en.wikipedia.org/wiki/Mersenne_prime#List_of_known_Mersenne_primes </big>
:: 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<sup>2147483647</sup> - 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'''.
<lang>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.
</lang>
[[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
Pari/GP solution for 2..11213 -> 2.835 s --[[User:Nebulus|Nebulus]] ([[User talk:Nebulus|talk]]) 05:21, 22 February 2023 (UTC)
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