mlucas - program to perform Lucas-Lehmer test on a Mersenne number, 2 ^ p - 1
mlucas -s tiny | t | small | s | medium | m | large | l | huge | h | all | a [-iters 100 | 1000 | 10000 [-nthread threads]]
mlucas -m exponent | -f exponent [-iters 100 | 1000 | 10000 [-nthread threads]]
mlucas -fftlen fft_length [-radset radix_set] [-m exponent | -f exponent] -iters 100 | 1000 | 10000 [-nthread threads]
This manual page documents briefly the mlucas command.
mlucas is an open-source (and free/libre) program for performing Lucas-Lehmer test on prime-exponent Mersenne numbers, that is, integers of the form 2 ^ p - 1, with prime exponent p. In short, everything you need to search for world-record Mersenne primes! It has been used in the verification of various Mersenne primes, including the 45th, 46th and 48th found Mersenne prime.
You may use it to test any suitable number as you wish, but it is preferable that you do so in a coordinated fashion, as part of the Great Internet Mersenne Prime Search (GIMPS). For more information on GIMPS, see the Great Internet Mersenne Prime Search subsection within the NOTES section and SEE ALSO section. Note that mlucas is not (yet) as efficient as the main GIMPS client, George Woltman’s Prime95 program (a.k.a. mprime for the (gnu/)linux version), but that program is not truly open-source (and free/libre), since it requires the user to abide by the prize-sharing rules set by its author (incompatible with freedom to run the program as you wish, for any purpose), should a user be lucky enough to find a new prime eligible for one of the monetary prizes offered by the Electronic Freedom Foundation (see EFF Cooperative Computing Awards <https://www.eff.org/awards/coop> for details).
mlucas reads the exponents from the
$MLUCAS_PATH/worktodo.ini file. Results are written to the
$MLUCAS_PATH/results.txt file and the exponent-specific
$MLUCAS_PATH/*.stat file (see section FILES for details). Error messages are written to stderr and the
$MLUCAS_PATH/*.stat file. Exponents can also be passed as command-line arguments but this is mainly used for debugging (see section OPTIONS for details). In addition, mlucas can perform the Pe’pin primality test on Fermat numbers 2 ^ (2 ^ n) + 1, using an exponent-optimized fast-transform length much like that used for testing Mersenne numbers.
New users are urged to jump straight to the EXAMPLE section and follow the examples and pointers to other sections. Users with little time for in-depth reading should at least read the NOTES, BUGS and EXAMPLE sections for a brief introduction to the Great Internet Mersenne Prime Search, undesirable restrictions and common usages. FILES section is also highly recommended since it describes the mlucas configuration files used for host-specific optimization and other mlucas-generated files. Advanced users should also peruse the OPTIONS section since it introduces less-commonly-used advanced options. Experienced users who find this manual inadequate should consult the SEE ALSO section for further information. Lastly, the Mlucas README, available both online and offline, is highly recommended since it is written and maintained by the author of mlucas and should be considered the final authority.
mlucas follows the traditional POSIX (see standards(7) for details) command line syntax, with short options starting with one dashes (‘-’). A summary of options is included below. A complete description is in the SEE ALSO section.
|-h||Show version of program and summary of options.|
|-s t, -s tiny||Run 100-iteration self-test on a set of 32 Mersenne exponents, ranging from 173431 to 2455003. This will take around 1 minute on a fast (pre-2010) CPU.|
|-s s, -s small|
|Run 100-iteration self-test on a set of 24 Mersenne exponents, ranging from 173431 to 1245877. This will take around 10 minutes on a fast (pre-2010) CPU.|
|-s m, -s medium|
|Run 100-iteration self-test on a set of 24 Mersenne exponents, ranging from 1327099 to 9530803. This will take around an hour on a fast (pre-2010) CPU.|
|-s l, -s large|
|Run 100-iteration self-test on a set of 24 Mersenne exponents, ranging from 10151971 to 72851621. This will take around an hour on a fast (pre-2010) CPU.|
|-s h, -s huge||Run 100-iteration self-test on a set of 16 Mersenne exponents, ranging from 77597293 to 282508657. This will take a couple of hours on a fast (pre-2010) CPU.|
|-s a, -s all||Run 100-iteration self-test on all Mersenne exponents and all FFT radix sets. This will take several hours on a fast (pre-2010) CPU.|
|This allows the user to specify the length of the fast-transform (FFT) used to effect the large-integer modular multiply which is at the heart of all such nonfactorial primality tests. The length unit here is in terms of the number of double-precision machine words used in the multiword-integer encoding of the primality test residue which is both input and result of each of said multiplies. Because mlucas is intended for testing numbers with many millions of bits, we generally speak of these FFT lengths in terms of kilodoubles (= 2 ^ 10 or 1024 doubles). If fft_length is one of the available FFT lengths (in kilodoubles), run all available FFT radices available at that length, unless the -radset flag is also invoked (see below for details). If -fftlen is invoked with either the -m or -f flag, the self-tests will perform the first 100 iterations of a Lucas-Lehmer test (-m) or Pe’pin test (-f) on the user-specified Mersenne or Fermat number. If no user-set exponent is invoked, do 100 Lucas-Lehmer test iterations using the default self-test Mersenne or Fermat exponent for that FFT length. The program uses this to find the optimal radix set for a given FFT length on your hardware.|
|-iters 100 | 1000 | 10000|
|Do 100, 1000 or 10000 self-test iterations of the type determined by the modulus-related options (-s / -m = Lucas-Lehmer test iterations with initial seed 4, -f = Pe’pin test squarings with initial seed 3). Default is 100 iterations.|
|Specify index of a set of complex FFT radices to use, based on the big selection table in the function get_fft_radices(). This requires a supported value of -fftlen to be specified, meaning (for an FFT length supported by the program) an index 0, 1, 2, ... and so on. 0 is always a valid radix set index; how high one can go in the enumeration depends on the FFT length. As soon as the user tries an index out of range of the current FFT length, the program will error-exit with an informational message to that effect, which also notes the maximum allowable radix set index for that FFT length.|
|For multithread-enabled (default) build, perform the test in parallel mode with this many threads.|
|Perform a Lucas-Lehmer primality test of the Mersenne number M(exponent) = 2 ^ exponent - 1, where exponent must be an odd prime. If -iters is also invoked, this indicates a timing test. This requires suitable added arguments (-fftlen and, optionally, -radset) to be supplied. If the -fftlen option (and optionally -radset) is also invoked but -iters is not, the program first checks the first line of the
|Perform a base-3 Pe’pin test on the Fermat number F(exponent) = 2 ^ (2 ^ exponent) + 1. If desired this can be invoked together with the -fftlen option as for the Mersenne-number self-tests (see above notes on the -m flag; note that not all FFT lengths supported for -m are available for -f: -m permits FFT lengths of form odd * 2 ^ n with odd = any of 1, 3, 5, 7, 9, 11, 13, 15; -f allows odd = 1, 7, 15 and 63) Optimal radix sets and timings are written to the
|The list of exit status values is limited. It is not possible to determine the cause of failure from the exit status value alone. However, mlucas make use of stderr to print error messages as well as saving them to the
|for Mersenne number 2 ^ exponent - 1 or|
Cannot determine the number of CPUs.
Unknown fetal error.
Radix set index not available for given FFT length.
malloc(3), calloc(3) or realloc(3) failure.
pthread_create(3) or pthread_join(3) failure.
mlucas honors the following environment variables, if they exist:
|The path to read mlucas configuration files and to write mlucas generated files (see FILES section for details). MLUCAS_PATH must end with a slash (e.g., /home/foolish/bar/. If MLUCAS_PATH is not set, then MLUCAS_PATH defaults to
This section details mlucas configuration files and mlucas generated files. As noted in the ENVIRONMENT section,
$HOME/mlucas.d/ but this can be overridden at run-time by setting the MLUCAS_PATH environment variable.
|The format of this file is exactly the same as the format of
Normally, the timing entry for each line should be monotonic from above to below since larger FFT length should take longer to test. But it is OK for a given fft_length to have a higher timing than the one after it since mlucas checks the timings listed in this file for all FFT lengths >= the default FFT length for the number being tested, and uses the FFT length having the smallest listed timing. However, if you notice that this file has any entries such that a given fft_length has a timing 5% or more greater than the next-larger FFT length, or higher timing than two or more larger FFT lengths, please contact the author (see BUGS section for details).
|This file sets the number of threads used. It should only contain a positive integer since the content of this file is read by sscanf(in_line, %d, &NTHREADS); where the variable in_line contains the content of the
The assignment field contains Test if the assignment is a first-time Lucas-Lehmer test, or DoubleCheck if the assignment is a double-check Lucas-Lehmer test. (The program handles both cases the same way.)
ID is a unique 32-digit hex number.
exponent specifies the Mersenne number (of the form 2 ^ exponent - 1) to be tested.
trial factored up to is the number of bit this Mersenne number has been trial factored up to without finding a factor.
has P-1 factoring = 0 if no prior P-1 factoring has been done, = 1 if P-1 factoring (without finding a factor) has been done. Since mlucas currently has no P-1 factoring capability it simply discards these data, but users should prefer = 1 here since such an assignment is slightly more likely (5-10%) to yield a prime.
To do Lucas-Lehmer test, you should reserve exponents from the PrimeNet server and copy lines in the above format into the
|Save files in
|All files matching the following extended regular expression (see regex(7) for details) in
For both of the supported test types, duplicate pairs of savefiles are written at each checkpoint, to guard against corruption of the on-disk savefiles. Lucas-Lehmer test savefile-pair names start with <p> and <q>, respectively, while Pe’pin test savefile-pair names start with <f> and <q>, respectively. They should not be modified but backups may be made by the user. By default, the program will save a persistent backup of the primary (p or f) save file every 10 millionth iteration, for examples upon completion of the Lucas-Lehmer test of M57885161 the user will find the following exponent-associated files in the
Great Internet Mersenne Prime Search
This subsection needs to be compeleted...
The argument parser is buggy. The relative position of arguments is relevant to mlucas, the order of arguments in SYNOPSIS should be followed to avoid confusing the parser. Only 100, 1000 and 10000 are supported for -iters flag. However, the parser will not reject unsupported arguments. Using unsupported arguments for -iters flag may trigger strange behaviour.
Sometimes there is more than one applicable exit status values (see EXIT STATUS section for details). In such case, there is no guarantee which will be returned. For example, if malloc(3) failure triggers an assertion failure. It is possible that mlucas returns 1 instead of 255 as exit status value.
There are 3 common cases where you will want to run this program. Normally, you should do a spot-check first to quick-test your build, followed by the self-test range for ‘medium’ exponents. Finally, full-blown Lucas-Lehmer testing which is the main purpose of this program.
|mlucas -fftlen 192 -iters 100 -radset 0 -nthread 2|
|Perform spot-check to see if mlucas works and fill-in a bug report if it does not. The spot check should produce residues matching the internal tabulated ones. If the residues does not match, mlucas should emit a verbose error message.|
|mlucas -s m|
|Perform timing self-test for ‘medium’ exponents to tune code parameters for your platform. Ordinary users are recommended to do this self-test only. For best results, run any self-tests under zero- or constant-load conditions. The self-tests append (or create if
Perform Lucas-Lehmer test on Mersenne numbers by running mlucas as a background job (see JOB CONTROL section in bash(1) and Builtins subsection in dash(1) for details). To perform Lucas-Lehmer test on a given Mersenne number, you must first perform a self-test for ‘medium’ exponents mentioned above, or if you only desire to test a single selected Mersenne number, a self-test for the default FFT length for that number:
mlucas -m exponent -iters 100
In the case of multi-exponent production testing, you should reserve exponent from the PrimeNet server and add them into
$MLUCAS_PATH/worktodo.ini (see the subsection Great Internet Mersenne Prime Search within the section NOTES and FILES section for details).
Advanced Usage Tips
To start mlucas in terminal 1, add the following lines to your login shell initialization file, such as
$HOME/.profile (see INVOCATION section in bash(1) and Invocation subsection dash(1) for details).
# Test if we are in tty1 if test `tty` = /dev/tty1 then # turn on job control set -m # start mlucas nice mlucas > /dev/null 2>&1 & fi
bash(1), dash(1), reportbug(1)
mlucas is documented fully by Mlucas README, available both online and offline as shown above.
Great Internet Mersenne Prime Search <https://www.mersenne.org/>
Mersenne Forum <https://www.mersenneforum.org/>
Chris Caldwell’s web page on Mersenne numbers <https://primes.utm.edu/mersenne/index.html>
Richard Crandall and Barry Fagin, Discrete Weighted Transforms and Large-Integer Arithmetic. <https://pdfs.semanticscholar.org/07c0/fae878fe9d6a117de08282802fb7b892bf2d.pdf>
Richard E. Crandall, Ernst W. Mayer, and Jason S. Papadopoulos, The Twenty-Fourth Fermat Number is Composite. <https://www.mersenneforum.org/mayer/F24.pdf>