Provided by: libmath-prime-util-gmp-perl_0.50-1_amd64 bug

NAME

       Math::Prime::Util::GMP - Utilities related to prime numbers and factoring, using GMP

VERSION

       Version 0.50

SYNOPSIS

         use Math::Prime::Util::GMP ':all';
         my $n = "115792089237316195423570985008687907853269984665640564039457584007913129639937";

         # This doesn't impact the operation of the module at all, but does let you
         # enter big number arguments directly as well as enter (e.g.): 2**2048 + 1.
         use bigint;

         # These return 0 for composite, 2 for prime, and 1 for probably prime
         # Numbers under 2^64 will return 0 or 2.
         # is_prob_prime does a BPSW primality test for numbers > 2^64
         # is_prime adds some MR tests and a quick test to try to prove the result
         # is_provable_prime will spend a lot of effort on proving primality

         say "$n is probably prime"    if is_prob_prime($n);
         say "$n is ", qw(composite prob_prime def_prime)[is_prime($n)];
         say "$n is definitely prime"  if is_provable_prime($n) == 2;

         # Miller-Rabin and strong Lucas-Selfridge pseudoprime tests
         say "$n is a prime or spsp-2/7/61" if is_strong_pseudoprime($n, 2, 7, 61);
         say "$n is a prime or slpsp"       if is_strong_lucas_pseudoprime($n);
         say "$n is a prime or eslpsp"      if is_extra_strong_lucas_pseudoprime($n);

         # Return array reference to primes in a range.
         my $aref = primes( 10 ** 200, 10 ** 200 + 10000 );

         $next = next_prime($n);    # next prime > n
         $prev = prev_prime($n);    # previous prime < n

         # Primorials and lcm
         say "23# is ", primorial(23);
         say "The product of the first 47 primes is ", pn_primorial(47);
         say "lcm(1..1000) is ", consecutive_integer_lcm(1000);

         # Find prime factors of big numbers
         @factors = factor(5465610891074107968111136514192945634873647594456118359804135903459867604844945580205745718497);

         # Finer control over factoring.
         # These stop after finding one factor or exceeding their limit.
         #                               # optional arguments o1, o2, ...
         @factors = trial_factor($n);    # test up to o1
         @factors = prho_factor($n);     # no more than o1 rounds
         @factors = pbrent_factor($n);   # no more than o1 rounds
         @factors = holf_factor($n);     # no more than o1 rounds
         @factors = squfof_factor($n);   # no more than o1 rounds
         @factors = pminus1_factor($n);  # o1 = smoothness limit, o2 = stage 2 limit
         @factors = ecm_factor($n);      # o1 = B1, o2 = # of curves
         @factors = qs_factor($n);       # (no arguments)

DESCRIPTION

       A module for number theory in Perl using GMP.  This includes primality tests, getting
       primes in a range, factoring, and more.

       While it certainly can be used directly, the main purpose of this module is for
       Math::Prime::Util.  That module will automatically load this one if it is installed,
       greatly speeding up many of its operations on big numbers.

       Inputs and outputs for big numbers are via strings, so you do not need to use a bigint
       package in your program.  However if you do use bigints, inputs will be converted
       internally so there is no need to convert before a call.  Output results are returned as
       either Perl scalars (for native-size) or strings (for bigints).  Math::Prime::Util tries
       to reconvert all strings back into the callers bigint type if possible, which makes it
       more convenient for calculations.

       The various "is_*_pseudoprime" tests are more appropriately called "is_*_probable_prime"
       or "is_*_prp".  They return 1 if the input is a probable prime based on their test.  The
       naming convention is historical and follows Pari, Math::Primality, and some other math
       packages.  The modern definition of pseudoprime is a composite that passes the test,
       rather than any number.

FUNCTIONS

   is_prob_prime
         my $prob_prime = is_prob_prime($n);
         # Returns 0 (composite), 2 (prime), or 1 (probably prime)

       Takes a positive number as input and returns back either 0 (composite), 2 (definitely
       prime), or 1 (probably prime).

       For inputs below "2^64" the test is deterministic, so the possible return values are 0
       (composite) or 2 (definitely prime).

       For inputs above "2^64", a probabilistic test is performed.  Only 0 (composite) and 1
       (probably prime) are returned.  The current implementation uses the Baillie-PSW (BPSW)
       test.  There is a possibility that composites may be returned marked prime, but since the
       test was published in 1980, not a single BPSW pseudoprime has been found, so it is
       extremely likely to be prime.  While we believe (Pomerance 1984) that an infinite number
       of counterexamples exist, there is a weak conjecture (Martin) that none exist under 10000
       digits.

       In more detail, we are using the extra-strong Lucas test (Grantham 2000) using the Baillie
       parameter selection method (see OEIS A217719).  Previous versions of this module used the
       strong Lucas test with Selfridge parameters, but the extra-strong version produces fewer
       pseudoprimes while running 1.2 - 1.5x faster.  It is slightly stronger than the test used
       in Pari <http://pari.math.u-bordeaux.fr/faq.html#primetest>.

   is_prime
         say "$n is prime!" if is_prime($n);

       Takes a positive number as input and returns back either 0 (composite), 2 (definitely
       prime), or 1 (probably prime).  Composites will act exactly like "is_prob_prime", as will
       numbers less than "2^64".  For numbers larger than "2^64", some additional tests are
       performed on probable primes to see if they can be proven by another means.

       This call walks the line between the performance of "is_prob_prime" and the certainty of
       "is_provable_prime".  Those calls may be more appropriate in some cases.  What this
       function does is give most of the performance of the former, while adding more certainty.
       For finer tuning of this tradeoff, you may instead use "is_prob_prime" followed by
       additional probable prime tests such as "miller_rabin_random" and/or
       "is_frobenius_underwood_pseudoprime".

       As with "is_prob_prime", a BPSW test is first performed.  This is deterministic for all
       64-bit numbers.  Next, if the number is a Proth or LLR form, then a proof is constructed.
       If the result is still "probably prime" and the input is smaller than the Sorenson/Webster
       (2015) deterministic Miller-Rabin limit (approximately 82 bits) then the 11 or 12 Miller-
       Rabin tests are performed and the result is confirmed.  For larger inputs that are still
       "probably prime" but under 200 bits, a quick BLS75 "n-1" primality proof is attempted.
       This is tuned to give up if the result cannot be quickly determined, and results in
       success rates of ~80% at 80 bits, ~30% at 128 bits, and ~13% at 160 bits.  Lastly, for
       results still "probably prime", an additional random-base Miller-Rabin test is performed.

       The result is that many numbers will return 2 (definitely prime), and the numbers that
       return 1 (probably prime) have gone through more tests than "is_prob_prime" while not
       taking too long.

       For cryptographic key generation, you may want even more testing for probable primes (NIST
       recommends a few more additional M-R tests than we perform).  The function
       "miller_rabin_random" is made for this.  Alternately, a different test such as
       "is_frobenius_underwood_pseudoprime" can be used.  Even better, use "is_provable_prime"
       which should be reasonably fast for sizes under 2048 bits.  Typically for key generation
       one wants random primes, and there are many functions for that.

   is_provable_prime
         say "$n is definitely prime!" if is_provable_prime($n) == 2;

       Takes a positive number as input and returns back either 0 (composite), 2 (definitely
       prime), or 1 (probably prime).  A great deal of effort is taken to return either 0 or 2
       for all numbers.

       The current method first uses BPSW to find composites and provide a deterministic answer
       for tiny numbers (under "2^64").  If no certificate is required, LLR and Proth tests can
       be run, and small numbers (under approximately "2^82") can be satisfied with a
       deterministic Miller-Rabin test.  If the result is still not determined, a quick  BLS75
       "n-1" test is attempted, followed by ECPP.

       The time required for primes of different input sizes on a circa-2009 workstation averages
       about "3ms" for 30-digits, "5ms" for 40-digit, "20ms" for 60-digit, "50ms" for 80-digit,
       "100ms" for 100-digit, "2s" for 200-digit, and 400-digit inputs about a minute.  Expect a
       lot of time variation for larger inputs.  You can see progress indication if verbose is
       turned on (some at level 1, and a lot at level 2).

       A certificate can be obtained along with the result using the
       "is_provable_prime_with_cert" method.  There is no appreciable extra performance cost for
       returning a certificate.

   is_provable_prime_with_cert
       Takes a positive number as input and returns back an array with two elements.  The result
       will be one of:

         (0, '')      The input is composite.

         (1, '')      The input is probably prime but we could not prove it.
                      This is a failure in our ability to factor some necessary
                      element in a reasonable time, not a significant proof
                      failure (in other words, it remains a probable prime).

         (2, '...')   The input is prime, and the certificate contains all the
                      information necessary to verify this.

       The certificate is a text representation containing all the necessary information to
       verify the primality of the input in a reasonable time.  The result can be used with
       "verify_prime" in Math::Prime::Util for verification.  Proof types used include:

         ECPP
         BLS3
         BLS15
         BLS5
         Small

   is_pseudoprime
       Takes a positive number "n" and one or more non-zero positive bases as input.  Returns 1
       if the input is a probable prime to each base, 0 if not.  This is the simple Fermat
       primality test.  Removing primes, given base 2 this produces the sequence OEIS A001567
       <http://oeis.org/A001567>.

   is_euler_pseudoprime
       Takes a positive number "n" and one or more non-zero positive bases as input.  Returns 1
       if the input is an Euler probable prime to each base, 0 if not.  This is the Euler test,
       sometimes called the Euler-Jacobi test.  Removing primes, given base 2 this produces the
       sequence OEIS A047713 <http://oeis.org/A047713>.

   is_strong_pseudoprime
         my $maybe_prime = is_strong_pseudoprime($n, 2);
         my $probably_prime = is_strong_pseudoprime($n, 2, 3, 5, 7, 11, 13, 17);

       Takes a positive number "n" and one or more non-zero positive bases as input.  Returns 1
       if the input is a strong probable prime to each base, 0 if not.  This is often called the
       Miller-Rabin test.

       If 0 is returned, then the number really is a composite.  If 1 is returned, then it is
       either a prime or a strong pseudoprime to all the given bases.  Given enough distinct
       bases, the chances become very strong that the number is actually prime.

       Both the input number and the bases may be big integers.  If base modulo n <= 1 or base
       modulo n = n-1, then the result will be 1.  This allows the bases to be larger than n if
       desired, while still returning meaningful results.  For example,

         is_strong_pseudoprime(367, 1101)

       would incorrectly return 0 if this was not done properly.  A 0 result should be returned
       only if n is composite, regardless of the base.

       This is usually used in combination with other tests to make either stronger tests (e.g.
       the strong BPSW test) or deterministic results for numbers less than some verified limit
       (e.g. Jaeschke showed in 1993 that no more than three selected bases are required to give
       correct primality test results for any 32-bit number).  Given the small chances of passing
       multiple bases, there are some math packages that just use multiple MR tests for primality
       testing, though in the early 1990s almost all serious software switched to the BPSW test.

       Even numbers other than 2 will always return 0 (composite).  While the algorithm works
       with even input, most sources define it only on odd input.  Returning composite for all
       non-2 even input makes the function match most other implementations including
       Math::Primality's "is_strong_pseudoprime" function.

   miller_rabin_random
         my $maybe_prime = miller_rabin_random($n, 10); # 10 random bases

       Takes a positive number ("n") as input and a positive number ("k") of bases to use.
       Performs "k" Miller-Rabin tests using uniform random bases between 2 and "n-2".  This is
       the correct way to perform "k" Miller-Rabin tests, rather than the common but broken
       method of using the first "k" primes.

       An optional third argument may be given, which is a seed to use.  The seed should be a
       number either in decimal, binary with a leading "0b", hex with a leading "0x", or octal
       with a leading 0.  It will be converted to a GMP integer, so may be large.  Typically this
       is not necessary, but cryptographic applications may prefer the ability to use this, and
       it allows repeatable test results.

       There is no check for duplicate bases.  Input sizes below 65 bits make little sense for
       this function since is_prob_prime is deterministic at that size.  For numbers of 65+ bits,
       the chance of duplicate bases is quite small.  The exponentiation approximation for the
       birthday problem gives a probability of less than 2e-16 for 100 random bases to have a
       duplicate with a 65-bit input, and less than 2e-35 with a 128-bit input.

   is_lucas_pseudoprime
   is_strong_lucas_pseudoprime
       Takes a positive number as input, and returns 1 if the input is a standard or strong Lucas
       probable prime.  The Selfridge method of choosing D, P, and Q are used (some sources call
       this a Lucas-Selfridge test).  This is one half of the BPSW primality test (the Miller-
       Rabin strong probable prime test with base 2 being the other half).  The canonical BPSW
       test (page 1401 of Baillie and Wagstaff (1980)) uses the strong Lucas test with Selfridge
       parameters, but in practice a variety of Lucas tests with different parameters are used by
       tests calling themselves BPSW.

       The standard Lucas test implemented here corresponds to the Lucas test described in FIPS
       186-4 section C.3.3, though uses a slightly more efficient calculation.  Since the
       standard Lucas-Selfridge test is a subset of the strong Lucas-Selfridge test, I recommend
       using the strong test rather than the standard test for cryptographic purposes.  It is
       often slightly faster, has over 4x fewer pseudoprimes, and is the method recommended by
       Baillie and Wagstaff in their 1980 paper.

   is_extra_strong_lucas_pseudoprime
       Takes a positive number as input, and returns 1 if the input is an extra-strong Lucas
       probable prime.  This is defined in Grantham (2000), and is a slightly more stringent test
       than the strong Lucas test, though because different parameters are used the pseudoprimes
       are not a subset.  As expected by the extra conditions, the number of pseudoprimes is less
       than 2/3 that of the strong Lucas-Selfridge test.  Runtime performance is 1.2 to 1.5x
       faster than the strong Lucas test.

       The parameters are selected using the Baillie-OEIS method:

         P = 3;
         Q = 1;
         while ( jacobi( P*P-4, n ) != -1 )
           P += 1;

   is_almost_extra_strong_lucas_pseudoprime
       Takes a positive number as input and returns 1 if the input is an "almost" extra-strong
       Lucas probable prime.  This is the classic extra-strong Lucas test but without calculating
       the U sequence.  This makes it very fast, although as the input increases in size the time
       converges to the conventional extra-strong implementation:  at 30 digits this routine is
       about 15% faster, at 300 digits it is only 2% faster.

       With the current implementations, there is little reason to prefer this unless trying to
       reproduce specific results.  The extra-strong implementation has been optimized to use
       similar features, removing most of the performance advantage.

       An optional second argument (must be between 1 and 256) indicates the increment amount for
       P parameter selection.  The default value of one yields the method described in
       "is_extra_strong_lucas_pseudoprime".  A value of 2 yields the method used in Pari
       <http://pari.math.u-bordeaux.fr/faq.html#primetest>.

       Because the "U = 0" condition is ignored, this produces about 5% more pseudoprimes than
       the extra-strong Lucas test.  However this is still only 66% of the number produced by the
       strong Lucas-Selfridge test.  No BPSW counterexamples have been found with any of the
       Lucas tests described.

   is_euler_plumb_pseudoprime
       Takes a positive number "n" as input and returns 1 if "n" passes Colin Plumb's Euler
       Criterion primality test.  Pseudoprimes to this test are a subset of the base 2 Fermat and
       Euler tests, but a superset of the base 2 strong pseudoprime (Miller-Rabin) test.

       The main reason for this test is that is a bit more efficient than other probable prime
       tests.

   is_perrin_pseudoprime
       Takes a positive number "n" as input and returns 1 if "n" divides P(n) where P(n) is the
       Perrin number of "n".  The Perrin sequence is defined by "P(n) = P(n-2) + P(n-3)" with
       "P(0) = 3, P(1) = 0, P(2) = 2".

       This is not a commonly used test, as it runs slower than most of the other probable prime
       tests and offers little benefit, especially over combined tests like "is_bpsw_prime",
       "is_frobenius_underwood_pseudoprime", and "is_frobenius_khashin_pseudoprime".

       An optional second argument "r" indicates whether to run additional tests.  With "r=1",
       "P(-n) = -1 mod n" is also verified, creating the "minimal restricted" test.  With "r=2",
       the full signature is also tested using the Adams and Shanks (1982) rules (without the
       quadratic form test).  With "r=3", the full signature is tested using the Grantham (2000)
       test, which additionally does not allow pseudoprimes to be divisible by 2 or 23.  The
       minimal restricted pseudoprime sequence is OEIS A018187 <http://oeis.org/A018187>.

   is_frobenius_pseudoprime
       Takes a positive number "n" as input, and two optional parameters "a" and "b", and returns
       1 if the "n" is a Frobenius probable prime with respect to the polynomial "x^2 - ax + b".
       Without the parameters, "b = 2" and "a" is the least positive odd number such that
       "(a^2-4b|n) = -1".  This selection has no pseudoprimes below "2^64" and none known.  In
       any case, the discriminant "a^2-4b" must not be a perfect square.

   is_frobenius_underwood_pseudoprime
       Takes a positive number as input, and returns 1 if the input passes the efficient
       Frobenius test of Paul Underwood.  This selects a parameter "a" as the least non-negative
       integer such that "(a^2-4|n)=-1", then verifies that "(x+2)^(n+1) = 2a + 5 mod
       (x^2-ax+1,n)".  This combines a Fermat and Lucas test at a computational cost of about
       2.5x a strong pseudoprime test.  This makes it similar to, but faster than, a standard
       Frobenius test.

       This test is deterministic (no randomness is used).  There are no known pseudoprimes to
       this test.  This test also has no overlap with the BPSW test, making it a very effective
       method for adding additional certainty.

   is_frobenius_khashin_pseudoprime
       Takes a positive number as input, and returns 1 if the input passes the Frobenius test of
       Sergey Khashin.  This ensures "n" is not a perfect square, selects the parameter "c" as
       the smallest odd prime such that "(c|n)=-1", then verifies that "(1+D)^n = (1-D) mod n"
       where "D = sqrt(c) mod n".

       This test is deterministic (no randomness is used).  There are no known pseudoprimes to
       this test.

   is_bpsw_prime
       Given a positive number input, returns 0 (composite), 2 (definitely prime), or 1 (probably
       prime), using the BPSW primality test (extra-strong variant).

       This function does the extra-strong BPSW test and nothing more.  That is, it will skip all
       pretests and any extra work that the "is_prob_prime" test may add.  This saves some time
       if the input has no small factors, such as testing results that have been sieved.

   is_aks_prime
         say "$n is definitely prime" if is_aks_prime($n);

       Takes a positive number as input, and returns 1 if the input passes the Agrawal-Kayal-
       Saxena (AKS) primality test.  This is a deterministic unconditional primality test which
       runs in polynomial time for general input.

       The particular method used is theorem 4.1 from Bernstein (2003).  This is substantially
       faster than the original AKS publication, the later version with improvements by Lenstra
       (sometimes called the V6 paper), or the later improvements of Voloch and Bornemann.  It
       is, by a large order, faster than any other known implementation as of early 2017.

       For theoretical analysis of the primality task, AKS is extremely important.  In practice,
       it is essentially useless.  Estimated run time for a 150 digit input is over 2 days,
       making the case that while the algorithmic complexity growth is polynomial, the constants
       are extremely high.  It will take years for for numbers that ECPP or APR-CL can prove in
       seconds.

       With the "verbose" option set to 1, the chosen "r" and "s" values are printed before the
       test starts.  With "verbose" set to 2 or higher, each of the "s" tests results in a "."
       output as the test runs, allowing progress to be monitored.

       Typically you should use "is_provable_prime" and let it decide the method.

   is_mersenne_prime
         say "2^607-1 (M607) is a Mersenne prime" if is_mersenne_prime(607);

       Takes a positive number "p" as input and returns 1 if "2^p-1" is prime.  After some pre-
       testing, the Lucas-Lehmer test is performed.  This is a deterministic unconditional test
       that runs very fast compared to other primality methods for numbers of comparable size,
       and vastly faster than any known general-form primality proof methods.

   is_llr_prime
       Takes a positive number "n" as input and returns one of: 0 (definitely composite), 2
       (definitely prime), or -1 (test does not indicate anything).  This implements the Lucas-
       Lehmer-Riesel test for fast deterministic primality testing on numbers of the form "k *
       2^n - 1".  If the input is not of this form or if "k >= 2^n" then "-1" will be returned as
       the test does not apply.  If "k = 1" then this is a Mersenne number and the Lucas-Lehmer
       test is used.  Otherwise, the LLR test is performed.  While not as fast as the Lucas-
       Lehmer test for Mersenne numbers, it is almost as fast as a single strong pseudoprime test
       (i.e.  Miller-Rabin test) while giving a certain answer.

   is_proth_prime
       Takes a positive number "n" as input and returns one of: 0 (definitely composite), 2
       (definitely prime), or -1 (test does not indicate anything).  This applies Proth's theorem
       for fast Las Vegas primality testing on numbers of the form "k * 2^n + 1".  If the input
       is not of this form or if "k >= 2^n" then "-1" will be returned as the test does not
       apply.  Otherwise, a search is performed to find a quadratic nonresidue modulo "n".  If
       none can be found after a brief search, "-1" is returned as no conclusion can be reached.
       Otherwise, Proth's theorem is checked which conclusively indicates primality.  While not
       as fast as the Lucas-Lehmer test for Mersenne numbers, it is almost as fast as a single
       strong pseudoprime test (i.e.  Miller-Rabin test) while giving a certain answer.

   is_miller_prime
         say "$n is definitely prime" if is_miller_prime($n);
         say "$n is definitely prime assuming the GRH" if is_miller_prime($n, 1);

       Takes a positive number as input, and returns 1 if the input passes the deterministic
       Miller test.  An optional second argument indicates whether the Generalized Riemann
       Hypothesis should be assumed, and defaults to 0.  Setting the verbose flag to 2 or higher
       will show how many bases are used.  The unconditional test is exponential time, while the
       conditional test (assuming the GRH) is polynomial time.

       This is a very slow method in practice, and generally should not be used.  The asymptotic
       complexity of the GRH version is good in theory, matching ECPP, but in practice it is much
       slower.  The number of bases used by the unconditional test grows quite rapidly,
       impractically many past about 160 bits, and overflows a 64-bit integer at 456 bits --
       sizes that are trivial for the unconditional APR-CL and ECPP tests.

   is_nminus1_prime
         say "$n is definitely prime" if is_nminus1_prime($n);

       Takes a positive number as input, and returns 1 if the input passes either theorem 5 or
       theorem 7 of the Brillhart-Lehmer-Selfridge primality test.  This is a deterministic
       unconditional primality test which requires factoring "n-1" to a linear factor less than
       the cube root of the input.  For small inputs (under 40 digits) this is typically very
       easy, and some numbers will naturally lead to this being very fast.  As the input grows,
       this method slows down rapidly.

       This method is most appropriate for numbers of the form "k+1" where "k" can be easily
       factored.  Typically you should use "is_provable_prime" and let it decide the method.

   is_nplus1_prime
       Takes a positive number as input, and returns 1 if the input passes either theorem 17 or
       theorem 19 of the Brillhart-Lehmer-Selfridge primality test.  This is a deterministic
       unconditional primality test which requires factoring "n+1" to a linear factor less than
       the cube root of the input.  For small inputs (under 40 digits) this is typically very
       easy, and some numbers will naturally lead to this being very fast.  As the input grows,
       this method slows down rapidly.

       Disregarding factoring, this is slightly slower than the "n-1" methods.  It is most
       appropriate for numbers of the form "k-1" where "k" can be easily factored.

   is_bls75_prime
       Takes a positive number as input, and returns 1 if the input passes one of the tests from
       the Brillhart-Lehmer-Selfridge (1975) paper.  These use partial factoring of "n-1" and
       "n+1".  Currently the implementation will use one of:

         N-1   Corollary 1, Theorem 5, Theorem 7
         N+1   Corollary 8, Theorem 17, Theorem 19
         Comb  Theorem 20

       This is appropriate for cases where either "n-1" or "n+1" can be easily factored, or when
       both of them have many small factors.

   is_ecpp_prime
         say "$n is definitely prime" if is_ecpp_prime($n);

       Takes a positive number as input, and returns 1 if the input passes the ECPP primality
       test.  This is the Atkin-Morain Elliptic Curve Primality Proving algorithm.  It is the
       fastest primality proving method in Math::Prime::Util.

       This implementation uses a "factor all strategy" (FAS) with backtracking.  A limited set
       of about 500 precalculated discriminants are used, which works well for inputs up to 300
       digits, and for many inputs up to one thousand digits.  Having a larger set will help with
       large numbers (a set of 2650 is available on github in the "xt/" directory).  A future
       implementation may include code to generate class polynomials as needed.

       Typically you should use "is_provable_prime" and let it decide the method.

   primes
         my $aref1 = primes( 1_000_000 );
         my $aref2 = primes( 2 ** 448, 2 ** 448 + 10000 );
         say join ",", @{primes( 2**2048, 2**2048 + 10000 )};

       Returns all the primes between the lower and upper limits (inclusive), with a lower limit
       of 2 if none is given.

       An array reference is returned, matching the signature of the function of the same name in
       Math::Prime::Util.

       Values above 64-bit are extra-strong BPSW probable primes.

   prime_count
       Returns the number of primes between 2 and "n" (single argument) or "lo" and "hi" given
       two arguments.  The values are inclusive.

       The method is simple sieving followed by primality testing.  This is appropriate for small
       ranges and is useful for very large arguments.  The Math::Prime::Util module has much more
       sophisticated methods for 64-bit arguments.

   prime_count_lower
   prime_count_upper
       Returns lower or upper bounds for the prime count of the input "n".

       Bounds use Dusart 2010, Büthe 2014, Büthe 2015, and Axler 2017.

   sieve_primes
         my @primes = sieve_primes(2**100, 2**100 + 10000);
         my @candidates = sieve_primes(2**1000, 2**1000 + 10000, 40000);

       Given two arguments "low" and "high", this returns the primes in the interval (inclusive)
       as a list.  It operates similar to primes, though must always have an lower and upper
       bound and returns a list.

       With three arguments "low", "high", and "limit", this does a partial sieve over the
       inclusive range and returns the list that pass the sieve.  If "limit" is less than 2 then
       it is identical to the two-argument version, in that a primality test will be performed
       after sieving.  Otherwise, sieving is performed up to "limit".

       The two-argument version is typically only used internally and adds little functionality.
       The three-argument version is quite useful for applications that want to apply their own
       primality or other tests, and wish to have a list of values in the range with no small
       factors.  This is quite common for applications involving prime gaps.

       Also see "sieve_range".

   sieve_range
         my @candidates = sieve_range(2**1000, 10000, 40000);

       Given a start value "n", and native unsigned integers "width" and "depth", a sieve of
       maximum depth "depth" is done for the "width" consecutive numbers beginning with "n".  An
       array of offsets from the start is returned.

       The returned list contains those offsets in the range "n" to "n+width-1" where "n +
       offset" has no prime factors less than "depth".

       This function is very similar to the three argument form of "sieve_primes".  The
       differences are using "(n,width)" instead of "(low,high)", and most importantly returning
       small offsets from the start value rather than the values themselves.  This can
       substantially reduce overhead for multi-thousand digit numbers.

   sieve_twin_primes
         my @primes = sieve_twin_primes(2**1000, 2**1000 + 500000);

       Given two arguments "low" and "high", this returns each lower twin prime in the interval
       (inclusive).  The result is a list, not a reference.

       This does a partial sieve of the range, removes any non-twin candidates, then checks that
       each pair are both BPSW probable primes.  This is substantially more efficient than
       sieving for all primes followed by removing those that are not twin primes.

   sieve_prime_cluster
         # Find some prime septuplets
         my @s = sieve_prime_cluster(2**100, 2**100+1e12, 2,6,8,12,18,20);

       Efficiently finds prime clusters between the first two arguments "low" and "high"
       (inclusive).  The remaining arguments describe the cluster.  The cluster values must be
       even, less than 31 bits, and strictly increasing.  Given a cluster set "C", the returned
       values are all primes in the range where "p+c" is prime for all "c" in the cluster set
       "C".

       The cluster is described as offsets from 0, with the implicit prime at 0.  Hence an empty
       list is asking for all primes (the cluster "p+0").  A list with the single value 2 will
       find all twin primes (the cluster where "p+0" and "p+2" are prime).  The list "2,6,8" will
       find prime quadruplets.  Note that there is no requirement that the list denote a
       constellation (a cluster with minimal distance) -- the list "42,92,606" is just fine.

       For long clusters, e.g. OEIS series A213601 <http://oeis.org/A213601> prime 12-tuplets,
       this will be immensely more efficient than filtering out the cluster from a list of
       primes.  For that example, a range of "10^13" takes less than a second to search --
       thousands of times faster than filtering results from primes or twin primes.  Shorter
       clusters are not quite this efficient, and the overhead for returning large arrays should
       not be ignored.

   next_prime
         $n = next_prime($n);

       Returns the prime following the input number (the smallest prime number that is greater
       than the input number).  The function "is_prob_prime" is used to determine when a prime is
       found, hence the result is a probable prime (using BPSW).

       For large inputs this function is quite a bit faster than GMP's "mpz_nextprime" or Pari's
       "nextprime".

   prev_prime
         $n = prev_prime($n);

       Returns the prime preceding the input number (the largest prime number that is less than
       the input number).  0 is returned if the input is 2 or lower.  The function
       "is_prob_prime" is used to determine when a prime is found, hence the result is a probable
       prime (using BPSW).

   surround_primes
         ($dprev, $dnext) = surround_primes($n);

       Returns the distances to the previous and next primes of the input "n".  This is slightly
       more efficient than calling both "prev_prime" and "next_prime", and returning the
       distances as native integers is more efficient with large inputs.

       If an optional second argument "d" is given, and the input "n" is larger than "2^64", then
       if a SPSP-2 is found in the range "n-d" to "n+d" (inclusive) then it will be returned with
       the other argument set to 0.  Otherwise, the first SPSP-2 values found are returned.  This
       feature is especially useful for prime gap searches as well as finding the nearest prime
       to a value.

       Note that with a non-zero second argument, the values returned have not undergone a full
       BPSW test; just sieving and a SPSP-2 test.

   random_nbit_prime
         say "random 512-bit prime: ", random_nbit_prime(512);

       Returns a randomly selected prime of exactly "n" bits.  "undef" is returned if "n" is less
       than 2.  The returned prime has passed the "is_prob_prime" (extra strong BPSW) test.

   random_strong_prime
         say "random 512-bit strong prime: ", random_strong_prime(512);

       Returns a randomly selected strong prime of exactly "n" bits.  "n" must be at least 256.
       The returned prime has passed the "is_prob_prime" (extra strong BPSW) test.

       Given the returned prime "p", "p+1", "q=p-1", and "q-1" will all have a large factor.
       This makes using factoring methods such as p-1 and p+1 much harder.  Gordon's algorithm is
       used.  The value of using strong primes is questionable over proper random primes when the
       number of bits is at least 1024.

   random_ndigit_prime
         say "random 200-digit prime: ", random_ndigit_prime(200);

       Returns a randomly selected prime of exactly "n" digits.  "undef" is returned if "n" is
       less than 1.  The returned prime has passed the "is_prob_prime" (extra strong BPSW) test.

   random_prime
         say random_prime(1000, 2000);  # prime between 1000 and 2000 inclusive

       Returns a random prime in the interval "[a,b]" or "undef" if no prime is in the range.
       The returned prime has passed the "is_prob_prime" (extra strong BPSW) test.

       The random prime functions use the internal CSPRNG for randomness.  This is currently
       ISAAC-32 but will likely change to ChaCha20 in a later release.

       This corresponds to Mathematica's "RandomPrime[{min,max}]" function.  This is a superset
       of Pari's randomprime(n) function, where our interval API is more convenient for
       cryptographic functions.

   random_maurer_prime
         say "random 512-bit proven prime: ", random_maurer_prime(512);

       Returns an n-bit proven prime using Ueli Maurer's FastPrime algorithm (1995).  This
       results in uniform random selection of a proven prime, though not every n-bit prime can be
       generated with this algorithm.

       "undef" is returned if "n" is less than 2.  As a safety check, internally the extra strong
       BPSW test is additionally run on each intermediate and the final result.

   random_shawe_taylor_prime
         say "random 512-bit proven prime: ", random_shawe_taylor_prime(512);

       Returns an n-bit proven prime using the Shawe-Taylor algorithm (1986) from section C.6 of
       FIPS 186-4, although using our CSPRNG rather than SHA-256.  This is a slightly simpler and
       older (1986) method than Maurer's algorithm.  It is a bit faster than Maurer's method but
       has a smaller subset of returned primes.

       "undef" is returned if "n" is less than 2.  As a safety check, internally the extra strong
       BPSW test is additionally run on each intermediate and the final result.

   random_maurer_prime_with_cert
       Like "random_maurer_prime" but also returns a string certificate.

   random_shawe_taylor_prime_with_cert
       Like "random_shawe_taylor_prime" but also returns a string certificate.

   lucasu
         say "Fibonacci($_) = ", lucasu(1,-1,$_) for 0..100;

       Given integers "P", "Q", and the non-negative integer "k", computes "U_k" for the Lucas
       sequence defined by "P","Q".  These include the Fibonacci numbers ("1,-1"), the Pell
       numbers ("2,-1"), the Jacobsthal numbers ("1,-2"), the Mersenne numbers ("3,2"), and more.

       This corresponds to OpenPFGW's "lucasU" function and gmpy2's "lucasu" function.

   lucasv
         say "Lucas($_) = ", lucasv(1,-1,$_) for 0..100;

       Given integers "P", "Q", and the non-negative integer "k", computes "V_k" for the Lucas
       sequence defined by "P","Q".  These include the Lucas numbers ("1,-1").

       This corresponds to OpenPFGW's "lucasV" function and gmpy2's "lucasv" function.

   lucas_sequence
         my($U, $V, $Qk) = lucas_sequence($n, $P, $Q, $k)

       Computes "U_k", "V_k", and "Q_k" for the Lucas sequence defined by "P","Q", modulo "n".
       The modular Lucas sequence is used in a number of primality tests and proofs.

       The following conditions must hold:
         - "D = P*P - 4*Q  !=  0"
         - "P > 0"
         - "P < n"
         - "Q < n"
         - "k >= 0"
         - "n >= 2"

   primorial
         $p = primorial($n);

       Given an unsigned integer argument, returns the product of the prime numbers which are
       less than or equal to "n".  This definition of "n#" follows OEIS series A034386
       <http://oeis.org/A034386> and Wikipedia: Primorial definition for natural numbers
       <http://en.wikipedia.org/wiki/Primorial#Definition_for_natural_numbers>.

   pn_primorial
         $p = pn_primorial($n)

       Given an unsigned integer argument, returns the product of the first "n" prime numbers.
       This definition of "p_n#" follows OEIS series A002110 <http://oeis.org/A002110> and
       Wikipedia: Primorial definition for prime numbers
       <http://en.wikipedia.org/wiki/Primorial#Definition_for_prime_numbers>.

       The two are related with the relationships:

         pn_primorial($n)  ==   primorial( nth_prime($n) )
         primorial($n)     ==   pn_primorial( prime_count($n) )

   factorial
       Given positive integer argument "n", returns the factorial of "n", defined as the product
       of the integers 1 to "n" with the special case of "factorial(0) = 1".  This corresponds to
       Pari's factorial(n) and Mathematica's "Factorial[n]" functions.

   factorialmod
       Given two positive integer arguments "n" and "m", returns "n! mod m".  This is much faster
       than computing the large factorial(n) followed by a mod operation.

   gcd
       Given a list of integers, returns the greatest common divisor.  This is often used to test
       for coprimality <https://oeis.org/wiki/Coprimality>.

   lcm
       Given a list of integers, returns the least common multiple.

   gcdext
       Given two integers "x" and "y", returns "u,v,d" such that "d = gcd(x,y)" and "u*x + v*y =
       d".  This uses the extended Euclidian algorithm to compute the values satisfying Bézout's
       Identity.

       This corresponds to Pari's "gcdext" function, which was renamed from "bezout" in Pari 2.6.
       The results will hence match "bezout" in Math::Pari.

   chinese
         say chinese( [14,643], [254,419], [87,733] );  # 87041638

       Solves a system of simultaneous congruences using the Chinese Remainder Theorem (with
       extension to non-coprime moduli).  A list of "[a,n]" pairs are taken as input, each
       representing an equation "x ≡ a mod n".  If no solution exists, "undef" is returned.  If a
       solution is returned, the modulus is equal to the lcm of all the given moduli (see "lcm".
       In the standard case where all values of "n" are coprime, this is just the product.  The
       "n" values must be positive integers, while the "a" values are integers.

   vecsum
       Returns the sum of all arguments, each of which must be an integer.

   vecprod
       Returns the product of all arguments, each of which must be an integer.

   kronecker
       Returns the Kronecker symbol "(a|n)" for two integers.  The possible return values with
       their meanings for odd positive "n" are:

          0   a = 0 mod n
          1   a is a quadratic residue modulo n (a = x^2 mod n for some x)
         -1   a is a quadratic non-residue modulo n

       The Kronecker symbol is an extension of the Jacobi symbol to all integer values of "n"
       from the latter's domain of positive odd values of "n".  The Jacobi symbol is itself an
       extension of the Legendre symbol, which is only defined for odd prime values of "n".  This
       corresponds to Pari's "kronecker(a,n)" function and Mathematica's "KroneckerSymbol[n,m]"
       function.

   binomial
       Given integer arguments "n" and "k", returns the binomial coefficient
       "n*(n-1)*...*(n-k+1)/k!", also known as the choose function.  Negative arguments use the
       Kronenburg extensions <http://arxiv.org/abs/1105.3689/>.  This corresponds to
       Mathematica's "Binomial[n,k]" function, Pari's "binomial(n,k)" function, and GMP's
       "mpz_bin_ui" function.

       For negative arguments, this matches Mathematica.  Pari does not implement the "n < 0, k
       <= n" extension and instead returns 0 for this case.  GMP's API does not allow negative
       "k" but otherwise matches.  Math::BigInt does not implement any extensions and the results
       for "n < 0, k > 0" are undefined.

   logreal
       Returns the natural logarithm of the input "n".  An optional second argument indicates the
       number of significant digits (default 40) with the result rounded.

       For logreal(2) we use Formula 25 from Gourdon and Sebah (2010).  For other values we use
       AGM (Sasaki and Kanada theta method).  Performance is 100-1000x faster than
       Math::BigFloat's GMP backend.  It is 10x slower than Pari/GP 2.10 and MPFR.

       Negative inputs are returned as "-log(-n)", which matches bignum.  Pari/GP and Mathematica
       return "log(-n) + Pi*i".

   expreal
       Returns "e^n" for the input "n".  An optional second argument indicates the number of
       significant digits (default 40) with the result rounded.

       The implementation computes sinh(n), then "e^x" from that.

   powreal
       Returns "n^x" for the inputs "n" and "x".  An optional third argument indicates the number
       of significant digits (default 40) with the result rounded.

       Like logreal and expreal, this is a basic math function that is not available from the GMP
       library but implemented in MPFR.  Since the latter is not always available, this can be
       useful to have.

   agmreal
       Returns the Arithmetic-Geometric mean (AGM) of "a" and "b".  An optional third argument
       indicates the number of significant digits (default 40) with the result rounded.

       Examples of use include elementary constants (e.g. "Pi" and "e"), logs, exponentials,
       trigonometric functions, elliptic integrals, computing pendulum periods, and more.

       This corresponds to Pari's "agm(x,y)" function, limited to positive reals (Pari also
       handles negative, complex, p-adic, and power series arguments).

   bernfrac
       Returns the Bernoulli number "B_n" for an integer argument "n", as a rational number.  Two
       values are returned, the numerator and denominator.  B_1 = 1/2.  This corresponds to
       Pari's bernfrac(n) and Mathematica's "BernoulliB" functions.

   bernreal
       Returns the Bernoulli number "B_n" for an integer argument "n", as a string floating
       point.  An optional second argument indicates the number of significant digits to be used,
       with the result rounded.  The default is 40 digits.  This corresponds to Pari's "bernreal"
       function and.

   harmfrac
       Returns the Harmonic number "H_n" for an integer argument "n", as a rational number.  Two
       values are returned, the numerator and denominator.  numbers are the sum of reciprocals of
       the first "n" natural numbers: "1 + 1/2 + 1/3 + ... + 1/n".  This corresponds to
       Mathematica's "HarmonicNumber" function.

   harmreal
       Returns the Harmonic number "H_n" for an integer argument "n", as a string floating point.
       An optional second argument indicates the number of digits to be preserved past the
       decimal place, with a default of 40.

   stirling
         say "s(14,2) = ", stirling(14, 2);
         say "S(14,2) = ", stirling(14, 2, 2);

       Returns the Stirling numbers of either the first kind (default), the second kind, or the
       third kind (the unsigned Lah numbers), with the kind selected as an optional third
       argument.  It takes two non-negative integer arguments "n" and "k" plus the optional
       "type".  This corresponds to Pari's "stirling(n,k,{type})" function and Mathematica's
       "StirlingS1" / "StirlingS2" functions.

       Stirling numbers of the first kind are "-1^(n-k)" times the number of permutations of "n"
       symbols with exactly "k" cycles.  Stirling numbers of the second kind are the number of
       ways to partition a set of "n" elements into "k" non-empty subsets.  The Lah numbers are
       the number of ways to split a set of "n" elements into "k" non-empty lists.

   zeta
       Given a positive integer or float "n", returns the real Riemann Zeta value as a string
       floating point.  An optional second argument indicates the number of digits past the
       decimal point (default 40).

       The implementation is algorithm 2 of Borwein (1991).  Performance with integer inputs is
       good, but floating point arguments with high precision will be slower than methods using
       MPFR.  Math::Prime::Util will try to use Math::MPFR if possible.

   li
       Given a positive integer or float "n", returns the real Logarithmic Integral as a string
       floating point.  An optional second argument indicates the number of significant digits
       (default 40) with the result rounded.

       The implementation uses Ramanjan's series.  This corresponds to Mathematica's "Li"
       function.

   ei
       Given a positive integer or float "n", returns the real Exponential Integral as a string
       floating point.  An optional second argument indicates the number of significant digits
       (default 40) with the result rounded.

       The implementation is simply li(exp(x)).

   riemannr
       Given a positive integer or float "n", returns the real Riemann R function as a string
       floating point.  An optional second argument indicates the number of significant digits
       (default 40) with the result rounded.

       The implementation is the standard Gram series.  This corresponds to Mathematica's
       "RiemannR" function.

   lambertw
       Given a float "x", returns the principal branch of the Lambert W function.  This solves
       for "W" in the equation "x = W*exp(W)".  The input must not be less than "-1/e".  This
       corresponds to Pari's "lambertw" function and Mathematica's "ProductLog" / "LambertW"
       function.

   znorder
         $order = znorder(17, "100000000000000000000000065");

       Given two positive integers "a" and "n", returns the multiplicative order of "a" modulo
       "n".  This is the smallest positive integer "k" such that "a^k ≡ 1 mod n".  Returns 1 if
       "a = 1".  Returns undef if "a = 0" or if "a" and "n" are not coprime, since no value will
       result in 1 mod n.  This corresponds to Pari's "znorder(Mod(a,n))" function and
       Mathematica's "MultiplicativeOrder[a,n]" function.

   znprimroot
       Given a positive integer "n", returns the smallest primitive root of "(Z/nZ)^*", or
       "undef" if no root exists.  A root exists when "euler_phi($n) == carmichael_lambda($n)",
       which will be true for all prime "n" and some composites.

       OEIS A033948 <http://oeis.org/A033948> is a sequence of integers where the primitive root
       exists, while OEIS A046145 <http://oeis.org/A046145> is a list of the smallest primitive
       roots, which is what this function produces.

   is_primitive_root
       Given two non-negative numbers "a" and "n", returns 1 if "a" is a primitive root modulo
       "n", and 0 if not.  If "a" is a primitive root, then euler_phi(n) is the smallest "e" for
       which "a^e = 1 mod n".

   is_semiprime
       Given a positive integer "n", returns 1 if "n" is a semiprime, 0 otherwise.  A semiprime
       is the product of exactly two primes.

       The boolean result is the same as "scalar(factor(n)) == 2", but this function performs
       shortcuts that can greatly speed up the operation.

   is_carmichael
       Given a positive integer "n", returns 1 if "n" is a Carmichael number, 0 otherwise.  These
       are composites that satisfy "b^(n-1) ≡ 1 mod n" for all "1 < b < n" relatively prime to
       "n".  Alternately Korselt's theorem says these are composites such that "n" is square-free
       and "p-1" divides "n-1" for all prime divisors "p" of "n".

       Inputs greater than 50 digits use a probabilistic test to avoid fully factoring the input.

   is_fundamental
       Given a positive integer "n", returns 1 if "n" is a fundamental discriminant, 0 otherwise.

   is_totient
       Given an integer "n", returns 1 if there exists an integer "x" where "euler_phi(x) == n".

   is_polygonal
       Given integers "x" and "s", return 1 if x is an s-gonal number, 0 otherwise.  "s" must be
       greater than 2.

   polygonal_nth
       Given integers "x" and "s", return N if "x" is the "N-th" s-gonal number, 0 otherwise.

   sigma
         say "Sum of divisors of $n:", sigma( $n );
         say "sigma_2($n) = ", sigma($n, 2);
         say "Number of divisors: sigma_0($n) = ", sigma($n, 0);

       This function takes a positive integer as input and returns the sum of its divisors,
       including 1 and itself.  An optional second argument "k" may be given, which will result
       in the sum of the "k-th" powers of the divisors to be returned.

       This is known as the sigma function (see Hardy and Wright section 16.7, or OEIS A000203).
       The API is identical to Pari/GP's "sigma" function.  This function is useful for
       calculating things like aliquot sums, abundant numbers, perfect numbers, etc.

   ramanujan_tau
       Takes a positive integer as input and returns the value of Ramanujan's tau function.  The
       result is a signed integer.  This corresponds to Mathematica's "RamanujanTau" function and
       Pari's "ramanujantau" function.

   valuation
         say "$n is divisible by 2 ", valuation($n,2), " times.";

       Given integers "n" and "k", returns the numbers of times "n" is divisible by "k".  This is
       a very limited version of the algebraic valuation meaning, just applied to integers.  This
       corresponds to Pari's "valuation" function.  0 is returned if "n" or "k" is one of the
       values "-1", 0, or 1.

   hammingweight
       Given an integer "n", returns the binary Hamming weight of abs(n).  This is also called
       the population count, and is the number of 1s in the binary representation.  This
       corresponds to Pari's "hammingweight" function for "t_INT" arguments.

   moebius
         say "$n is square free" if moebius($n) != 0;
         $sum += moebius($_) for (1..200); say "Mertens(200) = $sum";
         say "Mertens(2000) = ", vecsum(moebius(0,2000));

       Returns μ(n), the Möbius function (also known as the Moebius, Mobius, or MoebiusMu
       function) for an integer input.  This function is 1 if "n = 1", 0 if "n" is not square
       free (i.e. "n" has a repeated factor), and "-1^t" if "n" is a product of "t" distinct
       primes.  This is an important function in prime number theory.  Like SAGE, we define
       "moebius(0) = 0" for convenience.

       If called with two arguments, they define a range "low" to "high", and the function
       returns an array with the value of the Möbius function for every n from low to high
       inclusive.

   invmod
         say "The inverse of 42 mod 2017 = ", invmod(42,2017);

       Given two integers "a" and "n", return the inverse of "a" modulo "n".  If not defined,
       undef is returned.  If defined, then the return value multiplied by "a" equals 1 modulo
       "n".

   sqrtmod
       Given two integers "a" and "p", return the square root of "a" mod "p".  If no square root
       exists, undef is returned.  If defined, the return value "s" will always satisfy
       "mulmod(s,s,p) = a".

       If "p" is not a prime, it is possible no result will be returned even though a modular
       root exists.

       Only one root is returned, even though there are at least two.  In the case of "p" a prime
       and a return value "s", then both "+s mod n" and "-s mod n" are roots.  The least "s" will
       be returned.  In the case of composites, many roots may exist, but only one will be
       returned.

   addmod
       Given three integers "a", "b", and "n" where "a" and "n" are unsigned, return "(a+b) mod
       n".  This is particularly useful when dealing with numbers that are larger than a half-
       word but still native size.  No bigint package is needed and this can be 10-200x faster
       than using one.

   mulmod
       Given three integers "a", "b", and "n" where "a" and "n" are unsigned, return "(a*b) mod
       n".  This is particularly useful when "n" fits in a native integer.  No bigint package is
       needed and this can be 10-200x faster than using one.

   powmod
       Given three integers "a", "b", and "n" where "a" and "n" are unsigned, return "(a ** b)
       mod n".  Typically binary exponentiation is used, so the process is very efficient.  With
       native size inputs, no bigint library is needed.

   divmod
       Given three integers "a", "b", and "n" where "a" and "n" are unsigned, return "(a/b) mod
       n".  This is done as "(a * (1/b mod n)) mod n".  If no inverse of "b" mod "n" exists then
       undef if returned.

   consecutive_integer_lcm
         $lcm = consecutive_integer_lcm($n);

       Given an unsigned integer argument, returns the least common multiple of all integers from
       1 to "n".  This can be done by manipulation of the primes up to "n", resulting in much
       faster and memory-friendly results than using factorials.

   partitions
       Calculates the partition function p(n) for a non-negative integer input.  This is the
       number of ways of writing the integer n as a sum of positive integers, without
       restrictions.  This corresponds to Pari's "numbpart" function and Mathematica's
       "PartitionsP" function.  The values produced in order are OEIS series A000041
       <http://oeis.org/A000041>.

       This uses a combinatorial calculation, which means it will not be very fast compared to
       Pari, Mathematica, or FLINT which use the Rademacher formula using multi-precision
       floating point.  In 10 seconds, the pure Perl version can produce "partitions(10_000)"
       while with Math::Prime::Util::GMP it can do "partitions(220_000)".  In contrast, in about
       10 seconds Pari can solve "numbpart(22_000_000)".

       If you want the enumerated partitions, see "forpart" in Math::Prime::Util or
       Integer::Partition.  These are fast and memory efficient iterators, but not practical for
       producing the partition number for values over 100 or so.

   numtoperm
         @p = numtoperm(10,654321);  # @p=(1,8,2,7,6,5,3,4,9,0)

       Given a non-negative integer "n" and integer "k", return the rank "k" lexicographic
       permutation of "n" elements.  "k" will be interpreted as mod "n!".

   permtonum
         $k = permtonum([1,8,2,7,6,5,3,4,9,0]);  # $k = 654321

       Given an array reference containing integers from 0 to "n", returns the lexicographic
       permutation rank of the set.  This is the inverse of the "numtoperm" function.  All
       integers up to "n" must be present.  The result will be between 0 and "n!-1".

   Pi
       Takes a positive integer argument "n" and returns the constant Pi with that many digits
       (including the leading 3).  Rounding is performed.

       The implementation uses either AGM or Ramanujan/Chudnovsky with binary splitting,
       depending on the number of digits.  It is a little over 2x faster than MPFR, similar in
       speed to Pari/GP, but about 1.5x slower than Xue's Chudnovsky demo from the GMP web site.
       Specialized programs such as "y-cruncher" are even faster.

       Note there is a non-trivial amount of overhead in turning the result into a string, as
       well as even more if using the ntheory module which further converts the result into a
       Math::BigFloat object.

       Called in void context, this just calculates and caches the result.

   Euler
       Takes a positive integer argument "n" and returns Euler's constant with that many digits.
       Rounding is performed.

       The implementation is Brent-McMillan algorithm B, just like Pari/GP.  Performance is about
       3x faster than Pari/GP, but 2-10x slower than MPFR which uses binary splitting.

       Called in void context, this just calculates and caches the result.

   exp_mangoldt
         say "exp(lambda($_)) = ", exp_mangoldt($_) for 1 .. 100;

       Returns EXP(Λ(n)), the exponential of the Mangoldt function (also known as von Mangoldt's
       function) for an integer value.  The Mangoldt function is equal to log p if n is prime or
       a power of a prime, and 0 otherwise.  We return the exponential so all results are
       integers.  Hence the return value for "exp_mangoldt" is:

          p   if n = p^m for some prime p and integer m >= 1
          1   otherwise.

   totient
         say "The Euler totient of $n is ", totient($n);

       Returns φ(n), the Euler totient function (also called Euler's phi or phi function) for an
       integer value.  This is an arithmetic function which counts the number of positive
       integers less than or equal to "n" that are relatively prime to "n".  Given the definition
       used, "totient" will return 0 for all "n < 1".  This follows the logic used by SAGE.
       Mathematica and Pari return "totient(-n)" for "n < 0".  Mathematica returns 0 for "n = 0",
       Pari pre-2.6.2 raises and exception, and Pari 2.6.2 and newer returns 2.

   jordan_totient
         say "Jordan's totient J_$k($n) is ", jordan_totient($k, $n);

       Returns Jordan's totient function for a given integer value.  Jordan's totient is a
       generalization of Euler's totient, where
         "jordan_totient(1,$n) == euler_totient($n)" This counts the number of k-tuples less than
       or equal to n that form a coprime tuple with n.  As with "totient", 0 is returned for all
       "n < 1".  This function can be used to generate some other useful functions, such as the
       Dedekind psi function, where "psi(n) = J(2,n) / J(1,n)".

   carmichael_lambda
       Returns the Carmichael function (also called the reduced totient function, or Carmichael
       λ(n)) of a positive integer argument.  It is the smallest positive integer "m" such that
       "a^m = 1 mod n" for every integer "a" coprime to "n".  This is OEIS series A002322
       <http://oeis.org/A002322>.

   liouville
       Returns λ(n), the Liouville function for a non-negative integer input.  This is -1 raised
       to -(n) (the total number of prime factors).

   is_power
         say "$n is a perfect square" if is_power($n, 2);
         say "$n is a perfect cube" if is_power($n, 3);
         say "$n is a ", is_power($n), "-th power";

       Given a single positive integer input "n", returns k if "n = p^k" for some integer "p > 1,
       k > 1", and 0 otherwise.  The k returned is the largest possible.  This can be used in a
       boolean statement to determine if "n" is a perfect power.

       If given two arguments "n" and "k", returns 1 if "n" is a "k-th" power, and 0 otherwise.
       For example, if "k=2" then this detects perfect squares.

       This corresponds to Pari/GP's "ispower" function, with the limitations of only integer
       arguments and no third argument may be given to return the root.

   is_square
       Given a positive integer "n", returns 1 if "n" is a perfect square, 0 otherwise.  This is
       identical to "is_power(n,2)".

       This corresponds to Pari/GP's "issquare" function.

   is_prime_power
       Given an integer input "n", returns "k" if "n = p^k" for some prime p, and zero otherwise.

       This corresponds to Pari/GP's "isprimepower" function.

   sqrtint
       Returns the truncated integer part of the square root of "n".

       This corresponds to Pari/GP's "sqrtint" function.

   rootint
       Given "n" and "k", returns the truncated integer part of the "k-th" root of "n".

       This corresponds to Pari/GP's "sqrtnint" function.

   logint
       Given "n" and "b", returns the integer base-"b" logarithm of "n".  This is the largest
       integer "e" such that "b^e <= n".

       This corresponds to Pari/GP's "logint" function.

   factor
         @factors = factor(640552686568398413516426919223357728279912327120302109778516984973296910867431808451611740398561987580967216226094312377767778241368426651540749005659);
         # Returns an array of 11 factors

       Returns a list of prime factors of a positive number, in numerical order.  The special
       cases of "n = 0" and "n = 1" will return "n".

       Like most advanced factoring programs, a mix of methods is used.  This includes trial
       division for small factors, perfect power detection, Pollard's Rho, Pollard's P-1 with
       various smoothness and stage settings, Hart's OLF (a Fermat variant), ECM (elliptic curve
       method), and QS (quadratic sieve).  Certainly improvements could be designed for this
       algorithm (suggestions are welcome).

       In practice, this factors 26-digit semiprimes in under "100ms", 36-digit semiprimes in
       under one second.  Arbitrary integers are factored faster.  It is many orders of magnitude
       faster than any other factoring module on CPAN circa 2013.  It is comparable in speed to
       Math::Pari's "factorint" for most inputs.

       If you want better factoring in general, I recommend looking at the standalone programs
       yafu <http://sourceforge.net/projects/yafu/>, msieve
       <http://sourceforge.net/projects/msieve/>, gmp-ecm <http://ecm.gforge.inria.fr/>, and
       GGNFS <http://sourceforge.net/projects/ggnfs/>.

   divisors
         my @divisors = divisors(30);   # returns (1, 2, 3, 5, 6, 10, 15, 30)

       Produces all the divisors of a positive number input, including 1 and the input number.
       The divisors are a power set of multiplications of the prime factors, returned as a sorted
       list with no duplications.  The result is identical to that of Pari's "divisors" and
       Mathematica's "Divisors[n]" functions.

       In scalar context this returns the sigma0 function (OEIS A000005), and is the same result
       as evaluating the returned array in scalar context (but much more efficient).  The result
       then corresponds to Pari's "numdiv" and Mathematica's "DivisorSigma[0,n]" functions.

   trial_factor
         my @factors = trial_factor($n);
         my @factors = trial_factor($n, 1000);

       Given a positive number input, tries to discover a factor using trial division.  The
       resulting array will contain either two factors (it succeeded) or the original number (no
       factor was found).  In either case, multiplying @factors yields the original input.  An
       optional divisor limit may be given as the second parameter.  Factoring will stop when the
       input is a prime, one factor is found, or the input has been tested for divisibility with
       all primes less than or equal to the limit.  If no limit is given, then "2**31-1" will be
       used.

       This is a good and fast initial test, and will be very fast for small numbers (e.g. under
       1 million).  For larger numbers, faster methods for complete factoring have been known
       since the 17th century.

       For inputs larger than about 1000 digits, a dynamic product/remainder tree is used, which
       is faster than GMP's native methods.  This helps when pruning composites or looking for
       very small factors.

   prho_factor
         my @factors = prho_factor($n);
         my @factors = prho_factor($n, 100_000_000);

       Given a positive number input, tries to discover a factor using Pollard's Rho method.  The
       resulting array will contain either two factors (it succeeded) or the original number (no
       factor was found).  In either case, multiplying @factors yields the original input.  An
       optional number of rounds may be given as the second parameter.  Factoring will stop when
       the input is a prime, one factor has been found, or the number of rounds has been
       exceeded.

       This is the Pollard Rho method with "f = x^2 + 3" and default rounds 64M.  It is very good
       at finding small factors.  Typically "pbrent_factor" will be preferred as it behaves
       similarly but runs quite a bit faster.  They use different parameters however, so are not
       completely identical.

   pbrent_factor
         my @factors = pbrent_factor($n);
         my @factors = pbrent_factor($n, 100_000_000);

       Given a positive number input, tries to discover a factor using Pollard's Rho method with
       Brent's algorithm.  The resulting array will contain either two factors (it succeeded) or
       the original number (no factor was found).  In either case, multiplying @factors yields
       the original input.  An optional number of rounds may be given as the second parameter.
       Factoring will stop when the input is a prime, one factor has been found, or the number of
       rounds has been exceeded.

       This is the Pollard Rho method using Brent's modified cycle detection, delayed "gcd"
       computations, and backtracking.  It is essentially Algorithm P''2 from Brent (1980).
       Parameters used are "f = x^2 + 3" and default rounds 64M.  It is very good at finding
       small factors.

   pminus1_factor
         my @factors = pminus1_factor($n);

         # Set B1 smoothness to 10M, second stage automatically set.
         my @factors = pminus1_factor($n, 10_000_000);

         # Run p-1 with B1 = 10M, B2 = 100M.
         my @factors = pminus1_factor($n, 10_000_000, 100_000_000);

       Given a positive number input, tries to discover a factor using Pollard's "p-1" method.
       The resulting array will contain either two factors (it succeeded) or the original number
       (no factor was found).  In either case, multiplying @factors yields the original input.
       An optional first stage smoothness factor (B1) may be given as the second parameter.  This
       will be the smoothness limit B1 for the first stage, and will use "10*B1" for the second
       stage limit B2.  If a third parameter is given, it will be used as the second stage limit
       B2.  Factoring will stop when the input is a prime, one factor has been found, or the
       algorithm fails to find a factor with the given smoothness.

       This is Pollard's "p-1" method using a default smoothness of 5M and a second stage of "B2
       = 10 * B1".  It can quickly find a factor "p" of the input "n" if the number "p-1" factors
       into small primes.  For example "n = 22095311209999409685885162322219" has the factor "p =
       3916587618943361", where "p-1 = 2^7 * 5 * 47 * 59 * 3137 * 703499", so this method will
       find a factor in the first stage if "B1 >= 703499" or in the second stage if "B1 >= 3137"
       and "B2 >= 703499".

       The implementation is written from scratch using the basic algorithm including a second
       stage as described in Montgomery 1987.  It is faster than most simple implementations I
       have seen (many of which are written assuming native precision inputs), but slower than
       Ben Buhrow's code used in earlier versions of yafu
       <http://sourceforge.net/projects/yafu/>, and nowhere close to the speed of the version
       included with modern GMP-ECM with large B values (it is actually quite a bit faster than
       GMP-ECM with small smoothness values).

   pplus1_factor
         my @factors = pplus1_factor($n);

       Given a positive number input, tries to discover a factor using Williams' "p+1" method.
       The resulting array will contain either two factors (it succeeded) or the original number
       (no factor was found).  In either case, multiplying @factors yields the original input.
       An optional first stage smoothness factor (B1) may be given as the second parameter.  This
       will be the smoothness limit B1 for the first stage.  Factoring will stop when the input
       is a prime, one factor has been found, or the algorithm fails to find a factor with the
       given smoothness.

   holf_factor
         my @factors = holf_factor($n);
         my @factors = holf_factor($n, 100_000_000);

       Given a positive number input, tries to discover a factor using Hart's OLF method.  The
       resulting array will contain either two factors (it succeeded) or the original number (no
       factor was found).  In either case, multiplying @factors yields the original input.  An
       optional number of rounds may be given as the second parameter.  Factoring will stop when
       the input is a prime, one factor has been found, or the number of rounds has been
       exceeded.

       This is Hart's One Line Factorization method, which is a variant of Fermat's algorithm.  A
       premultiplier of 480 is used.  It is very good at factoring numbers that are close to
       perfect squares, or small numbers.  Very naive methods of picking RSA parameters sometimes
       yield numbers in this form, so it can be useful to run a few rounds to check.  For
       example, the number:

         18548676741817250104151622545580576823736636896432849057 \
         10984160646722888555430591384041316374473729421512365598 \
         29709849969346650897776687202384767704706338162219624578 \
         777915220190863619885201763980069247978050169295918863

       was proposed by someone as an RSA key.  It is indeed composed of two distinct prime
       numbers of similar bit length.  Most factoring methods will take a very long time to break
       this.  However one factor is almost exactly 5x larger than the other, allowing HOLF to
       factor this 222-digit semiprime in only a few milliseconds.

   squfof_factor
         my @factors = squfof_factor($n);
         my @factors = squfof_factor($n, 100_000_000);

       Given a positive number input, tries to discover a factor using Shanks' square forms
       factorization method (usually known as SQUFOF).  The resulting array will contain either
       two factors (it succeeded) or the original number (no factor was found).  In either case,
       multiplying @factors yields the original input.  An optional number of rounds may be given
       as the second parameter.  Factoring will stop when the input is a prime, one factor has
       been found, or the number of rounds has been exceeded.

       This is Daniel Shanks' SQUFOF (square forms factorization) algorithm.  The particular
       implementation is a non-racing multiple-multiplier version, based on code ideas of Ben
       Buhrow and Jason Papadopoulos as well as many others.  SQUFOF is often the preferred
       method for small numbers, and Math::Prime::Util as well as many other packages use it was
       the default method for native size (e.g. 32-bit or 64-bit) numbers after trial division.
       The GMP version used in this module will work for larger values, but my testing indicates
       it is generally slower than the "prho" and "pbrent" implementations.

   ecm_factor
         my @factors = ecm_factor($n);
         my @factors = ecm_factor($n, 12500);      # B1 = 12500
         my @factors = ecm_factor($n, 12500, 10);  # B1 = 12500, curves = 10

       Given a positive number input, tries to discover a factor using ECM.  The resulting array
       will contain either two factors (it succeeded) or the original number (no factor was
       found).  In either case, multiplying @factors yields the original input.  An optional
       maximum smoothness may be given as the second parameter, which relates to the size of
       factor to search for.  An optional third parameter indicates the number of random curves
       to use at each smoothness value being searched.

       This is an implementation of Hendrik Lenstra's elliptic curve factoring method, usually
       referred to as ECM.  The implementation is reasonable, using projective coordinates,
       Montgomery's PRAC heuristic for EC multiplication, and two stages.  It is much slower than
       the latest GMP-ECM, but still quite useful for factoring reasonably sized inputs.

   qs_factor
         my @factors = qs_factor($n);

       Given a positive number input, tries to discover factors using QS (the quadratic sieve).
       The resulting array will contain one or more numbers such that multiplying @factors yields
       the original input.  Typically multiple factors will be produced, unlike the other
       "..._factor" routines.

       The current implementation is a modified version of SIMPQS, a predecessor to the QS in
       FLINT, and was written by William Hart in 2006.  It will not operate on input less than 30
       digits.  The memory use for large inputs is more than desired, so other methods such as
       "pbrent_factor", "pminus1_factor", and "ecm_factor" are recommended to begin with to
       filter out small factors.  However, it is substantially faster than the other methods on
       large inputs having large factors, and is the method of choice for 35+ digit semiprimes.

   todigits
       Given an integer "n", return an array of digits of "|n|".  An optional second integer
       argument specifies a base (default 10).  For example, given a base of 2, this returns an
       array of binary digits of "n".  An optional third argument specifies a length for the
       returned array.  The result will be either have upper digits truncated or have leading
       zeros added.

       todigits(0) returns an empty array.  The base must be at least 2, and is limited to an
       int.  Length must be at least zero and is limited to an int.

   seed_csprng
       Takes a non-negative integer "nbytes" and a string "data" as input.  These are used to
       seed the internal CSPRNG used for random functions, including the random prime functions.
       Ideally this is 16-256 bytes of good entropy.

       Currently the CSPRNG is ISAAC-32, and the maximum number of seed bytes used is 1024.  The
       CSPRNG will likely change to ChaCha20 in a later release.

   is_csprng_well_seeded
       Returns true if the CSPRNG has been seeded with 16 or more bytes (128 bits).  There is no
       measurement of how "good" the input was.

       On startup the module will attempt to seed the CSPRNG from "/dev/urandom", so this
       function will return true if that was successful, but false otherwise.

   urandomb
         $n32 = urandomb(32);    # Classic irand32, returns a UV
         $n   = urandomb(1024);  # Random integer less than 2^1024

       Given a number of bits "b", returns a random unsigned integer less than "2^b".  The result
       will be uniformly distributed between 0 and "2^b-1" inclusive.

       This is similar to the GMP function "mpz_urandomb".

   urandomm
         $n = urandomm(100);    # random integer in [0,99]
         $n = urandomm(1024);   # random integer in [0,1023]

       Given a positive integer "n", returns a random unsigned integer less than "n".  The
       results will be uniformly distributed between 0 and "n-1" inclusive.

       This is similar to the GMP function "mpz_urandomm".

   urandomr
         $n  = urandomr(100, 110);        # Random number [100,110]
         $nb = urandomr(2**24,2**25-1);   # Random 25-bit number
         $nd = urandomr(10**24,10**25-1); # Random 25-digit number

       Given values "low" and "high", returns a uniform random unsigned integer in the range
       "[low,high]".  Both inputs must be non-negative.  If "low > high" then function will
       return "undef".  Note that the range is inclusive, so "low", "high", and each integer
       between them have an equal probability of appearing.

   irand
         $n32 = irand;     # random 32-bit integer

       Returns a random 32-bit integer using the CSPRNG.

       Performance is similar to "rand" in Math::Random::MTwist and Math::Random::Xorshift.  It
       is somewhat faster than casting system "rand" to a 32-bit int.  It is noticeably faster
       than Math::Random::ISAAC, Math::Random::ISAAC::XS, Math::Random::MT,
       Math::Random::MT::Auto, and Crypt::PRNG.

   irand64
         $n64 = irand64;   # random 64-bit integer

       Returns a random 64-bit integer using the CSPRNG (on 64-bit Perl).

   drand
         $f = drand;       # random floating point value in [0,1)
         $r = drand(25);   # random floating point value in [0,25)

       Returns a random NV (Perl's native floating point) using the CSPRNG.

       The number of bits returned is equal to the mantissa bits of the NV type used for the Perl
       build, with a max of 64.  By default Perl uses doubles and the returned values have 53
       bits.  If Perl is built with long double support and the long doubles have a larger
       mantissa, then more bits are used.

       This gives substantially better quality random numbers than the default Perl "rand"
       function.  Among other things, on modern Perl's, "rand" uses drand48, which gives 32 bits
       of decent random values and 16 more bits of known patterns (e.g. the 48th bit alternates,
       the 47th has a period of 4, etc.).  There are much better choices for standard random
       number generators, such as the Mersenne Twister from Math::Random::MTwist.

       Performance is similar to "rand" in Math::Random::MTwist and Math::Random::Xorshift.  It
       is 1.5 - 2x slower than core "rand" (as are the other modules).

   random_bytes
         $str = random_bytes(32);     # 32 random bytes

       Given an unsigned number "n" of bytes, returns a binary string filled with random data
       from the CSPRNG.  Performance for getting 256 byte strings:

           Module/Method                  Rate   Type
           -------------             ---------   ----------------------
           Data::Entropy::Algorithms    2027/s   CSPRNG - AES Counter
           Crypt::Random                6649/s   CSPRNG - /dev/urandom
           Bytes::Random                9217/s   drand48
           Bytes::Random::Secure       23043/s   CSPRNG - ISAAC
           Math::Random::ISAAC::XS     58377/s   CSPRNG - ISAAC
           rand+pack                   82977/s   drand48
           Crypt::PRNG                298567/s   CSPRNG - Fortuna
           Bytes::Random::XS          383354/s   drand48
           ntheory                    770364/s   CSPRNG - ChaCha20
           Math::Random::MTwist      1890151/s   Mersenne Twister
           Math::Prime::Util::GMP    2045715/s   CSPRNG - ISAAC

       Each of the CSPRNG modules should be high quality.  There are no known flaws in any of
       ISAAC, AES CTR, ChaCha20, or Fortuna.  The industry seems to be standardizing on ChaCha20
       (e.g. BSD, Linux, TLS).

SEE ALSO

       Math::Prime::Util Has many more functions, lots of fast code for dealing with native-
       precision arguments (including much faster primes using sieves), and will use this module
       when needed for big numbers.  Using Math::Prime::Util rather than this module directly is
       recommended.
       Math::Primality (version 0.08) A Perl module with support for the strong Miller-Rabin
       test, strong Lucas-Selfridge test, the BPSW probable prime test, next_prime / prev_prime,
       the AKS primality test, and prime_count.  It uses Math::GMPz to do all the calculations,
       so is faster than pure Perl bignums, but a lot slower than XS+GMP.  The prime_count
       function is only usable for very small inputs, but the other functions are quite good for
       big numbers.  Make sure to use version 0.05 or newer.
       Math::Pari Supports quite a bit of the same functionality (and much more).  See "SEE ALSO"
       in Math::Prime::Util for more detailed information on how the modules compare.
       yafu <http://sourceforge.net/projects/yafu/>, msieve
       <http://sourceforge.net/projects/msieve/>, gmp-ecm <http://ecm.gforge.inria.fr/>, GGNFS
       <http://sourceforge.net/projects/ggnfs/> Good general purpose factoring utilities.  These
       will be faster than this module, and much better as the factor increases in size.
       Primo <http://www.ellipsa.eu/public/primo/primo.html> is the state of the art in freely
       available (though not open source!) primality proving programs.  If you have 1000+ digit
       numbers to prove, you want to use this.
       mpz_aprcl <http://sourceforge.net/projects/mpzaprcl/> Open source APR-CL primality proof
       implementation. Fast primality proving, though without certificates.

REFERENCES

       Robert Baillie and Samuel S. Wagstaff, Jr., "Lucas Pseudoprimes", Mathematics of
       Computation, v35 n152, October 1980, pp 1391-1417.
       <http://mpqs.free.fr/LucasPseudoprimes.pdf>
       Daniel J. Bernstein, "Proving Primality After Agrawal-Kayal-Saxena", preprint, Jan 2003.
       <http://cr.yp.to/papers/aks.pdf>
       Jon Grantham, "Frobenius Pseudoprimes", Mathematics of Computation, v70 n234, March 2000,
       pp 873-891.  <http://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01197-2/>
       John Brillhart, D. H. Lehmer, and J. L. Selfridge, "New Primality Criteria and
       Factorizations of 2^m +/- 1", Mathematics of Computation, v29, n130, Apr 1975, pp 620-647.
       <http://www.ams.org/journals/mcom/1975-29-130/S0025-5718-1975-0384673-1/S0025-5718-1975-0384673-1.pdf>
       Richard P. Brent, "An improved Monte Carlo factorization algorithm", BIT 20, 1980, pp.
       176-184.  <http://www.cs.ox.ac.uk/people/richard.brent/pd/rpb051i.pdf>
       Peter L. Montgomery, "Speeding the Pollard and Elliptic Curve Methods of Factorization",
       Mathematics of Computation, v48, n177, Jan 1987, pp 243-264.
       <http://www.ams.org/journals/mcom/1987-48-177/S0025-5718-1987-0866113-7/>
       Richard P. Brent, "Parallel Algorithms for Integer Factorisation", in Number Theory and
       Cryptography, Cambridge University Press, 1990, pp 26-37.
       <http://www.cs.ox.ac.uk/people/richard.brent/pd/rpb115.pdf>
       Richard P. Brent, "Some Parallel Algorithms for Integer Factorisation", in Proc. Third
       Australian Supercomputer Conference, 1999. (Note: there are multiple versions of this
       paper)  <http://www.cs.ox.ac.uk/people/richard.brent/pd/rpb193.pdf>
       William B. Hart, "A One Line Factoring Algorithm", preprint.
       <http://wstein.org/home/wstein/www/home/wbhart/onelinefactor.pdf>
       Daniel Shanks, "SQUFOF notes", unpublished notes, transcribed by Stephen McMath.
       <http://www.usna.edu/Users/math/wdj/mcmath/shanks_squfof.pdf>
       Jason E. Gower and Samuel S. Wagstaff, Jr, "Square Form Factorization", Mathematics of
       Computation, v77, 2008, pages 551-588.  <http://homes.cerias.purdue.edu/~ssw/squfof.pdf>
       A.O.L. Atkin and F. Morain, "Elliptic Curves and primality proving", Mathematics of
       Computation, v61, 1993, pages 29-68.
       <http://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1199989-X/>
       R.G.E. Pinch, "Some Primality Testing Algorithms", June 1993.  Describes the primality
       testing methods used by many CAS systems and how most were compromised.  Gives
       recommendations for primality testing APIs.
       <http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.33.4409>

AUTHORS

       Dana Jacobsen <dana@acm.org>

       Jason Papadopoulos wrote the tinyqs code which is basically unchanged.  William Hart wrote
       SIMPQS which is the basis for the QS code.

ACKNOWLEDGEMENTS

       Obviously none of this would be possible without the mathematicians who created and
       published their work.  Eratosthenes, Gauss, Euler, Riemann, Fermat, Lucas, Baillie,
       Pollard, Brent, Montgomery, Shanks, Hart, Wagstaff, Dixon, Pomerance, A.K. Lenstra, H. W.
       Lenstra Jr., Atkin, Knuth, etc.

       The GNU GMP team, whose product allows me to concentrate on coding high-level algorithms
       and not worry about any of the details of how modular exponentiation and the like happen,
       and still get decent performance for my purposes.

       Ben Buhrow and Jason Papadopoulos deserve special mention for their open source factoring
       tools, which are both readable and fast.  In particular I am leveraging their work on
       SQUFOF in the current implementation.  They are a huge resource to the community.

       Jonathan Leto and Bob Kuo, who wrote and distributed the Math::Primality module on CPAN.
       Their implementation of BPSW provided the motivation I needed to do it in this module and
       Math::Prime::Util.  I also used their module quite a bit for testing against.

       Paul Zimmermann's papers and GMP-ECM code were of great value for my projective ECM
       implementation, as well as the many papers by Brent and Montgomery.

COPYRIGHT

       Copyright 2011-2017 by Dana Jacobsen <dana@acm.org>

       This program is free software; you can redistribute it and/or modify it under the same
       terms as Perl itself.

       SIMPQS Copyright 2006, William Hart.  SIMPQS is distributed under GPL v2+.