MLUCAS(1)

NAME
mlucas - program to perform Lucas-Lehmer test on a Mersenne number, 2 ^ p - 1

SYNOPSIS
mlucas
mlucas -h
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]

DESCRIPTION
[FIXME: This man page is potentially outdated, please see the online documentation <https://www.mersenneforum.org/mayer/README.html> or the help text located in /usr/share/doc/mlucas/help.txt.gz for up-to-date information.]

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.

OPTIONS
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.

-fftlen fft_length
    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.

-radset radix_set
    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.

-nthread threads
    For multithread-enabled (default) build, perform the test in parallel mode with this many threads.

-m exponent
    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 $MLUCAS_PATH/worktodo.ini file to see if the assignment specified there is a Lucas-Lehmer test with the same exponent as specified via the -m argument. If so, the -fftlen argument is treated as a user override of the default FFT length for the exponent. If -radset is also invoked, this is similarly treated as a user-specified radix set for the user-set FFT length; otherwise the program will use the $MLUCAS_PATH/mlucas.cfg file to select the radix set to be used for the user-forced FFT length. If the $MLUCAS_PATH/worktodo.ini file entry does not match the -m value, a set of timing self-tests is run on the user-specified Mersenne number using all sets of FFT radices available at the specified FFT length. If the -fftlen option is not invoked, the tests use all sets of FFT radices available at that exponent's default FFT length. Use this to find the optimal radix set for a single given Mersenne exponent on your hardware, similarly to the -fftlen option. Perform 100 iterations, or as many as specified via the -iters flag.

-f exponent
    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 $MLUCAS_PATH/fermat.cfg file. Perform 100 iterations, or as many as specified via the -iters flag.

EXIT STATUS

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 $MLUCAS_PATH/*.stat file, where * is in the form

    pexponent

for Mersenne number 2 ^ exponent - 1 or

    fexponent

for Fermat number 2 ^ (2 ^ exponent) + 1. (see FILES section for details).

0   Exit successfully.

1   Assertion failure.

    Cannot determine the number of CPUs.

    Unknown fetal error.

    Radix set index not available for given FFT length.

255 thread_policy_set() failure.

    malloc(3), calloc(3) or realloc(3) failure.

    pthread_create(3) or pthread_join(3) failure.

ENVIRONMENT
mlucas honors the following environment variables, if they exist:

MLUCAS_PATH
    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 $HOME/.mlucas.d/, where the environmental variable $HOME will be expanded in the environment where mlucas is invoked. mlucas will attept to make the directory with parents pointed by MLUCAS_PATH using the mkdir(1) command. The effect is similar to executing mkdir -p $MLUCAS_PATH in the shell provided that the -p flag is honored.

FILES
This section details mlucas configuration files and mlucas generated files. As noted in the ENVIRONMENT section, $MLUCAS_PATH defaults to $HOME/mlucas.d/ but this can be overridden at run-time by setting the MLUCAS_PATH environment variable.

$MLUCAS_PATH/*.stat
    The filename-prefix wildcard * is as described in the EXIT STATUS section; for the primality test of the Mersenne number 2 ^ exponent - 1 it is of the form

        pexponent

    All important events, per-10000-iteration residues (or per-100000-iteration if more than 4 threads are used for the test) and the final residue during Lucas-Lehmer test of exponent are recorded in this file. It can be seen as an exponent-specific detailed $MLUCAS_PATH/results.txt (see $MLUCAS_PATH/results.txt below for details). This file is useful for debugging purposes. Its format looks like:

        INFO: event

        ...

        Mexponent: using FFT length fft_lengthK = fft_length * 1024 8-byte floats. Bz<this gives an average> bits bits per digit

        Using complex FFT radices radix_set (product of all elements of radix_set = fft_length / 2)

        ...

        [date_and_time] Mexponent Iter# = iterations clocks = time_taken_per_10000_iterations [ time_taken_per_iteration sec/iter] Res64: residue. AvgMaxErr = roe_avg. MaxErr = roe_max

        ...

        [Restarting Mexponent at iteration = iteration. Res64: residue

        Mexponent: using FFT length fft_lengthK = fft_length * 1024 8-byte floats.

        this gives an average bits bits per digit

        Using complex FFT radices radix_set] (product of all elements of radix_set = fft_length / 2)

        ...

        Mexponent is not prime. Res64: residue. Program: E17.1

        Mexponent mod 2^36 = remainder_1

        Mexponent mod 2^35 - 1 = remainder_2

        Mexponent mod 2^36 - 1 = remainder_3

$MLUCAS_PATH/fermat.cfg
    The format of this file is exactly the same as the format of $MLUCAS_PATH/mlucas.cfg (see $MLUCAS_PATH/mlucas.cfg below for details).

$MLUCAS_PATH/mlucas.cfg
    This file stores the radix set with best timing for each FFT length. Its format looks like:

        17.1

        fft_length msec/iter = timing ROE[avg,max] = [roe_avg, roe_max] radices = radix_set

        ...

    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).

$MLUCAS_PATH/nthreads.ini
    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 $MLUCAS_PATH/nthreads.ini file. If this file is not present, mlucas will use as many threads as the number of CPUs detected. The number of threads used set by this file can be overridden by setting -nthread flag at run-time. This file is for those who want to set the number of threads to be greater or less than the number of CPUs detected. This can be useful since some users reported up to 10% performance gain when using more threads than the number of CPUs detected.

$MLUCAS_PATH/results.txt
    Important events which occurred during Lucas-Lehmer test and the final residue obtained are recorded in this file. This file summarizes important information in all $MLUCAS_PATH/*.stat files (see $MLUCAS_PATH/*.stat above for details) into a single file. This file (more specifically, any results which were added to it since your last checkin from) should be submitted to the PrimeNet server (see subsection Great Internet Mersenne Prime Search in section NOTES for details) since the Lucas-Lehmer test exponents are obtained from the PrimeNet server (see $MLUCAS_PATH/worktodo.ini below for details). Its format looks like:

        INFO: event

        ...

        [Mexponent Roundoff warning on iteration iteration, maxerr = roundoff_error

        Retrying iteration interval to see if roundoff error is reproducible.

        [Retry of iteration interval with fatal roundoff error was successful.]]

        ...

        Mexponent is not prime. Res64: residue. Program: E17.1

        Mexponent mod 2^36 = remainder_1

        Mexponent mod 2^35 - 1 = remainder_2

        Mexponent mod 2^36 - 1 = remainder_3

        ...

$MLUCAS_PATH/worktodo.ini
    This file contains Lucas-Lehmer test assignments to be tested. Its format looks like:

        assignment=ID,exponent,trial factored up to,has P-1 factoring

        ...

    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 $MLUCAS_PATH/worktodo.ini file (see subsection Great Internet Mersenne Prime Search in section NOTES for details). You may need to create the $MLUCAS_PATH/worktodo.ini file if it does not exist.

Save files in $MLUCAS_PATH
    All files matching the following extended regular expression (see regex(7) for details) in $MLUCAS_PATH directory are save files:

        ^[fpq][0123456789]+([.][0123456789]+0M)?$

    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 $MLUCAS_PATH directory:

        p57885161.stat
        p57885161.10M
        p57885161.20M
        p57885161.30M
        p57885161.40M
        p57885161.50M

NOTES

  Great Internet Mersenne Prime Search
This subsection needs to be compeleted...

BUGS
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.

For problems regarding the program mlucas, please contact the author Ernst W. Mayer <ewmayer AT aol DOT com>. For installation and documentation related problems regarding the Debian package and this manual, please use reportbug(1) to contact Alex Vong <alexvong1995 AT protonmail DOT com>.

EXAMPLE
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 $MLUCAS_PATH/mlucas.cfg does not exist) new timing data to the $MLUCAS_PATH/mlucas.cfg (see FILES section for details). Before doing any self-tests, you should first check if there is an existing $MLUCAS_PATH/mlucas.cfg file and either delete it or do a backup-via-rename to to prevent mixing old and new timing data. $MLUCAS_PATH/mlucas.cfg normally locates at $HOME/.mlucas.d/ directory although this can be overridden at run-time by settingthe MLUCAS_PATH environment variable (see ENVIRONMENT section for details).

mlucas &
    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

SEE ALSO
bash(1), dash(1), reportbug(1)

<https://www.mersenneforum.org/mayer/README.html>, /usr/share/doc/mlucas/html/README.html

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>