trusty (3) Math::Prime::Util.3pm.gz
NAME
Math::Prime::Util - Utilities related to prime numbers, including fast sieves and factoring
VERSION
Version 0.37
SYNOPSIS
# Normally you would just import the functions you are using.
# Nothing is exported by default. List the functions, or use :all.
use Math::Prime::Util ':all';
# Get a big array reference of many primes
my $aref = primes( 100_000_000 );
# All the primes between 5k and 10k inclusive
my $aref = primes( 5_000, 10_000 );
# If you want them in an array instead
my @primes = @{primes( 500 )};
# You can do something for every prime in a range. Twin primes to 10k:
forprimes { say if is_prime($_+2) } 10000;
# Or for the composites in a range
forcomposites { say if is_strong_pseudoprime($_,2) } 10000, 10**6;
# For non-bigints, is_prime and is_prob_prime will always be 0 or 2.
# They return 0 (composite), 2 (prime), or 1 (probably prime)
say "$n is prime" if is_prime($n);
say "$n is ", (qw(composite maybe_prime? prime))[is_prob_prime($n)];
# Strong pseudoprime test with multiple bases, using Miller-Rabin
say "$n is a prime or 2/7/61-psp" if is_strong_pseudoprime($n, 2, 7, 61);
# Standard and strong Lucas-Selfridge, and extra strong Lucas tests
say "$n is a prime or lpsp" if is_lucas_pseudoprime($n);
say "$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);
# step to the next prime (returns 0 if not using bigints and we'd overflow)
$n = next_prime($n);
# step back (returns 0 if given input less than 2)
$n = prev_prime($n);
# Return Pi(n) -- the number of primes E<lt>= n.
$primepi = prime_count( 1_000_000 );
$primepi = prime_count( 10**14, 10**14+1000 ); # also does ranges
# Quickly return an approximation to Pi(n)
my $approx_number_of_primes = prime_count_approx( 10**17 );
# Lower and upper bounds. lower <= Pi(n) <= upper for all n
die unless prime_count_lower($n) <= prime_count($n);
die unless prime_count_upper($n) >= prime_count($n);
# Return p_n, the nth prime
say "The ten thousandth prime is ", nth_prime(10_000);
# Return a quick approximation to the nth prime
say "The one trillionth prime is ~ ", nth_prime_approx(10**12);
# Lower and upper bounds. lower <= nth_prime(n) <= upper for all n
die unless nth_prime_lower($n) <= nth_prime($n);
die unless nth_prime_upper($n) >= nth_prime($n);
# Get the prime factors of a number
@prime_factors = factor( $n );
# Return ([p1,e1],[p2,e2], ...) for $n = p1^e1 * p2*e2 * ...
@pe = factor_exp( $n );
# Get all divisors other than 1 and n
@divisors = divisors( $n );
# Or just apply a block for each one
fordivisors { $sum += $_ + $_*$_ } $n;
# Euler phi (Euler's totient) on a large number
use bigint; say euler_phi( 801294088771394680000412 );
say jordan_totient(5, 1234); # Jordan's totient
# Moebius function used to calculate Mertens
$sum += moebius($_) for (1..200); say "Mertens(200) = $sum";
# Mertens function directly (more efficient for large values)
say mertens(10_000_000);
# Exponential of Mangoldt function
say "lamba(49) = ", log(exp_mangoldt(49));
# Some more number theoretical functions
say liouville(4292384);
say chebyshev_psi(234984);
say chebyshev_theta(92384234);
say partitions(1000);
# divisor sum
$sigma = divisor_sum( $n ); # sum of divisors
$sigma0 = divisor_sum( $n, 0 ); # count of divisors
$sigmak = divisor_sum( $n, $k );
$sigmaf = divisor_sum( $n, sub { log($_[0]) } ); # arbitrary func
# primorial n#, primorial p(n)#, and lcm
say "The product of primes below 47 is ", primorial(47);
say "The product of the first 47 primes is ", pn_primorial(47);
say "lcm(1..1000) is ", consecutive_integer_lcm(1000);
# Ei, li, and Riemann R functions
my $ei = ExponentialIntegral($x); # $x a real: $x != 0
my $li = LogarithmicIntegral($x); # $x a real: $x >= 0
my $R = RiemannR($x) # $x a real: $x > 0
my $Zeta = RiemannZeta($x) # $x a real: $x >= 0
# Precalculate a sieve, possibly speeding up later work.
prime_precalc( 1_000_000_000 );
# Free any memory used by the module.
prime_memfree;
# Alternate way to free. When this leaves scope, memory is freed.
my $mf = Math::Prime::Util::MemFree->new;
# Random primes
my $small_prime = random_prime(1000); # random prime <= limit
my $rand_prime = random_prime(100, 10000); # random prime within a range
my $rand_prime = random_ndigit_prime(6); # random 6-digit prime
my $rand_prime = random_nbit_prime(128); # random 128-bit prime
my $rand_prime = random_strong_prime(256); # random 256-bit strong prime
my $rand_prime = random_maurer_prime(256); # random 256-bit provable prime
DESCRIPTION
A set of utilities related to prime numbers. These include multiple sieving methods, is_prime,
prime_count, nth_prime, approximations and bounds for the prime_count and nth prime, next_prime and
prev_prime, factoring utilities, and more.
The default sieving and factoring are intended to be (and currently are) the fastest on CPAN, including
Math::Prime::XS, Math::Prime::FastSieve, Math::Factor::XS, Math::Prime::TiedArray, Math::Big::Factors,
Math::Factoring, and Math::Primality (when the GMP module is available). For numbers in the 10-20 digit
range, it is often orders of magnitude faster. Typically it is faster than Math::Pari for 64-bit
operations.
All operations support both Perl UV's (32-bit or 64-bit) and bignums. If you want high performance with
big numbers (larger than Perl's native 32-bit or 64-bit size), you should install Math::Prime::Util::GMP
and Math::BigInt::GMP. This will be a recurring theme throughout this documentation -- while all bignum
operations are supported in pure Perl, most methods will be much slower than the C+GMP alternative.
The module is thread-safe and allows concurrency between Perl threads while still sharing a prime cache.
It is not itself multi-threaded. See the Limitations section if you are using Win32 and threads in your
program.
Two scripts are also included and installed by default:
• primes.pl displays primes between start and end values or expressions, with many options for
filtering (e.g. twin, safe, circular, good, lucky, etc.). Use "--help" to see all the options.
• factor.pl operates similar to the GNU "factor" program. It supports bigint and expression inputs.
BIGNUM SUPPORT
By default all functions support bignums. For performance, you should install and use Math::BigInt::GMP
or Math::BigInt::Pari, and Math::Prime::Util::GMP.
If you are using bigints, here are some performance suggestions:
• Install Math::Prime::Util::GMP, as that will vastly increase the speed of many of the functions.
This does require the GMP <gttp://gmplib.org> library be installed on your system, but this
increasingly comes pre-installed or easily available using the OS vendor package installation tool.
• Install and use Math::BigInt::GMP or Math::BigInt::Pari, then use "use bigint try => 'GMP,Pari'" in
your script, or on the command line "-Mbigint=lib,GMP". Large modular exponentiation is much faster
using the GMP or Pari backends, as are the math and approximation functions when called with very
large inputs.
• Install Math::MPFR if you use the Ei, li, Zeta, or R functions. If that module can be loaded, these
functions will run much faster on bignum inputs, and are able to provide higher accuracy.
• I have run these functions on many versions of Perl, and my experience is that if you're using
anything older than Perl 5.14, I would recommend you upgrade if you are using bignums a lot. There
are some brittle behaviors on 5.12.4 and earlier with bignums. For example, the default BigInt
backend in older versions of Perl will sometimes convert small results to doubles, resulting in
corrupted output.
PRIMALITY TESTING
This module provides three functions for general primality testing, as well as numerous specialized
functions. The three main functions are: "is_prob_prime" and "is_prime" for general use, and
"is_provable_prime" for proofs. For inputs below "2^64" the functions are identical and fast
deterministic testing is performed. That is, the results will always be correct and should take at most
a few microseconds for any input. This is hundreds to thousands of times faster than other CPAN modules.
For inputs larger than "2^64", an extra-strong BPSW test <http://en.wikipedia.org/wiki/Baillie-
PSW_primality_test> is used. See the "PRIMALITY TESTING NOTES" section for more discussion.
FUNCTIONS
is_prime
print "$n is prime" if is_prime($n);
Returns 0 is the number is composite, 1 if it is probably prime, and 2 if it is definitely prime. For
numbers smaller than "2^64" it will only return 0 (composite) or 2 (definitely prime), as this range has
been exhaustively tested and has no counterexamples. For larger numbers, an extra-strong BPSW test is
used. If Math::Prime::Util::GMP is installed, some additional primality tests are also performed, and a
quick attempt is made to perform a primality proof, so it will return 2 for many other inputs.
Also see the "is_prob_prime" function, which will never do additional tests, and the "is_provable_prime"
function which will construct a proof that the input is number prime and returns 2 for almost all primes
(at the expense of speed).
For native precision numbers (anything smaller than "2^64", all three functions are identical and use a
deterministic set of tests (selected Miller-Rabin bases or BPSW). For larger inputs both "is_prob_prime"
and "is_prime" return probable prime results using the extra-strong Baillie-PSW test, which has had no
counterexample found since it was published in 1980.
For cryptographic key generation, you may want even more testing for probable primes (NIST recommends
some additional M-R tests). This can be done using a different test (e.g.
"is_frobenius_underwood_pseudoprime") or using additional M-R tests with random bases with
"miller_rabin_random". Even better, make sure Math::Prime::Util::GMP is installed and use
"is_provable_prime" which should be reasonably fast for sizes under 2048 bits. Another possibility is to
use "random_maurer_prime" in Math::Prime::Util which constructs a random provable prime.
primes
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 (with large lists this is much faster and uses less memory than returning
an array directly).
my $aref1 = primes( 1_000_000 );
my $aref2 = primes( 1_000_000_000_000, 1_000_000_001_000 );
my @primes = @{ primes( 500 ) };
print "$_\n" for @{primes(20,100)};
Sieving will be done if required. The algorithm used will depend on the range and whether a sieve result
already exists. Possibilities include primality testing (for very small ranges), a Sieve of Eratosthenes
using wheel factorization, or a segmented sieve.
next_prime
$n = next_prime($n);
Returns the next prime greater than the input number. The result will be a bigint if it can not be
exactly represented in the native int type (larger than "4,294,967,291" in 32-bit Perl; larger than
"18,446,744,073,709,551,557" in 64-bit).
prev_prime
$n = prev_prime($n);
Returns the prime preceding the input number (i.e. the largest prime that is strictly less than the
input). 0 is returned if the input is 2 or lower.
forprimes
forprimes { say } 100,200; # print primes from 100 to 200
$sum=0; forprimes { $sum += $_ } 100000; # sum primes to 100k
forprimes { say if is_prime($_+2) } 10000; # print twin primes to 10k
Given a block and either an end count or a start and end pair, calls the block for each prime in the
range. Compared to getting a big array of primes and iterating through it, this is more memory efficient
and perhaps more convenient. This will almost always be the fastest way to loop over a range of primes.
Nesting and use in threads are allowed.
Math::BigInt objects may be used for the range.
For some uses an iterator ("prime_iterator", "prime_iterator_object") or a tied array
(Math::Prime::Util::PrimeArray) may be more convenient. Objects can be passed to functions, and allow
early loop exits.
forcomposites
forcomposites { say } 1000;
forcomposites { say } 2000,2020;
Given a block and either an end number or a start and end pair, calls the block for each composite in the
inclusive range. The composites are the numbers greater than 1 which are not prime: "4, 6, 8, 9, 10, 12,
14, 15, ..."
fordivisors
fordivisors { $prod *= $_ } $n;
Given a block and a non-negative number "n", the block is called with $_ set to each divisor in sorted
order. Also see "divisor_sum".
prime_iterator
my $it = prime_iterator;
$sum += $it->() for 1..100000;
Returns a closure-style iterator. The start value defaults to the first prime (2) but an initial value
may be given as an argument, which will result in the first value returned being the next prime greater
than or equal to the argument. For example, this:
my $it = prime_iterator(200); say $it->(); say $it->();
will return 211 followed by 223, as those are the next primes >= 200. On each call, the iterator returns
the current value and increments to the next prime.
Other options include "forprimes" (more efficiency, less flexibility), Math::Prime::Util::PrimeIterator
(an iterator with more functionality), or Math::Prime::Util::PrimeArray (a tied array).
prime_iterator_object
my $it = prime_iterator_object;
while ($it->value < 100) { say $it->value; $it->next; }
$sum += $it->iterate for 1..100000;
Returns a Math::Prime::Util::PrimeIterator object. A shortcut that loads the package if needed, calls
new, and returns the object. See the documentation for that package for details. This object has more
features than the simple one above (e.g. the iterator is bi-directional), and also handles iterating
across bigints.
prime_count
my $primepi = prime_count( 1_000 );
my $pirange = prime_count( 1_000, 10_000 );
Returns the Prime Count function Pi(n), also called "primepi" in some math packages. When given two
arguments, it returns the inclusive count of primes between the ranges. E.g. "(13,17)" returns 2,
"(14,17)" and "(13,16)" return 1, "(14,16)" returns 0.
The current implementation decides based on the ranges whether to use a segmented sieve with a fast bit
count, or the extended LMO algorithm. The former is preferred for small sizes as well as small ranges.
The latter is much faster for large ranges.
The segmented sieve is very memory efficient and is quite fast even with large base values. Its
complexity is approximately "O(sqrt(a) + (b-a))", where the first term is typically negligible below "~
10^11". Memory use is proportional only to sqrt(a), with total memory use under 1MB for any base under
"10^14".
The extended LMO method has complexity approximately "O(b^(2/3)) + O(a^(2/3))", and also uses low memory.
A calculation of "Pi(10^14)" completes in a few seconds, "Pi(10^15)" in well under a minute, and
"Pi(10^16)" in about one minute. In contrast, even parallel primesieve would take over a week on a
similar machine to determine "Pi(10^16)".
Also see the function "prime_count_approx" which gives a very good approximation to the prime count, and
"prime_count_lower" and "prime_count_upper" which give tight bounds to the actual prime count. These
functions return quickly for any input, including bigints.
prime_count_upper
prime_count_lower
my $lower_limit = prime_count_lower($n);
my $upper_limit = prime_count_upper($n);
# $lower_limit <= prime_count(n) <= $upper_limit
Returns an upper or lower bound on the number of primes below the input number. These are analytical
routines, so will take a fixed amount of time and no memory. The actual "prime_count" will always be
equal to or between these numbers.
A common place these would be used is sizing an array to hold the first $n primes. It may be desirable
to use a bit more memory than is necessary, to avoid calling "prime_count".
These routines use verified tight limits below a range at least "2^35", and use the Dusart (2010) bounds
of
x/logx * (1 + 1/logx + 2.000/log^2x) <= Pi(x)
x/logx * (1 + 1/logx + 2.334/log^2x) >= Pi(x)
above that range. These bounds do not assume the Riemann Hypothesis. If the configuration option
"assume_rh" has been set (it is off by default), then the Schoenfeld (1976) bounds are used for large
values.
prime_count_approx
print "there are about ",
prime_count_approx( 10 ** 18 ),
" primes below one quintillion.\n";
Returns an approximation to the "prime_count" function, without having to generate any primes. For
values under "10^36" this uses the Riemann R function, which is quite accurate: an error of less than
"0.0005%" is typical for input values over "2^32", and decreases as the input gets larger. If Math::MPFR
is installed, the Riemann R function is used for all values, and will be very fast. If not, then values
of "10^36" and larger will use the approximation "li(x) - li(sqrt(x))/2". While not as accurate as the
Riemann R function, it still should have error less than "0.00000000000000001%".
A slightly faster but much less accurate answer can be obtained by averaging the upper and lower bounds.
nth_prime
say "The ten thousandth prime is ", nth_prime(10_000);
Returns the prime that lies in index "n" in the array of prime numbers. Put another way, this returns
the smallest "p" such that "Pi(p) >= n".
For relatively small inputs (below 1 million or so), this does a sieve over a range containing the nth
prime, then counts up to the number. This is fairly efficient in time and memory. For larger values,
create a low-biased estimate using the inverse logarithmic integral, use a fast prime count, then sieve
in the small difference.
While this method is thousands of times faster than generating primes, and doesn't involve big tables of
precomputed values, it still can take a fair amount of time for large inputs. Calculating the "10^12th"
prime takes about 1 second, the "10^13th" prime takes under 10 seconds, and the "10^14th" prime
(3475385758524527) takes under one minute. Think about whether a bound or approximation would be
acceptable, as they can be computed analytically.
If the result is larger than a native integer size (32-bit or 64-bit), the result will take a very long
time. A later version of Math::Prime::Util::GMP may include this functionality which would help for
32-bit machines.
nth_prime_upper
nth_prime_lower
my $lower_limit = nth_prime_lower($n);
my $upper_limit = nth_prime_upper($n);
# $lower_limit <= nth_prime(n) <= $upper_limit
Returns an analytical upper or lower bound on the Nth prime. These are very fast as they do not need to
sieve or search through primes or tables. An exact answer is returned for tiny values of "n". The lower
limit uses the Dusart 2010 bound for all "n", while the upper bound uses one of the two Dusart 2010
bounds for "n >= 178974", a Dusart 1999 bound for "n >= 39017", and a simple bound of "n * (logn + 0.6 *
loglogn)" for small "n".
nth_prime_approx
say "The one trillionth prime is ~ ", nth_prime_approx(10**12);
Returns an approximation to the "nth_prime" function, without having to generate any primes. Uses the
Cipolla 1902 approximation with two polynomials, plus a correction for small values to reduce the error.
is_pseudoprime
Takes a positive number "n" and a base "a" as input, and returns 1 if "n" is a probable prime to base
"a". This is the simple Fermat primality test. Removing primes, given base 2 this produces the sequence
OEIS A001567 <http://oeis.org/A001567>.
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 as input and one or more bases. The bases must be greater than 1. Returns 1 if
the input is a strong probable prime to all of the bases, and 0 if not.
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, very
strong that the number is actually prime.
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. it has long been known
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.
Even inputs other than 2 will always return 0 (composite). While the algorithm does run 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
An alias for "is_strong_pseudoprime". This name is deprecated.
is_lucas_pseudoprime
Takes a positive number as input, and returns 1 if the input is a standard Lucas probable prime using the
Selfridge method of choosing D, P, and Q (some sources call this a Lucas-Selfridge pseudoprime).
Removing primes, this produces the sequence OEIS A217120 <http://oeis.org/A217120>.
is_strong_lucas_pseudoprime
Takes a positive number as input, and returns 1 if the input is a strong Lucas probable prime using the
Selfridge method of choosing D, P, and Q (some sources call this a strong Lucas-Selfridge pseudoprime).
This is one half of the BPSW primality test (the Miller-Rabin strong pseudoprime test with base 2 being
the other half). Removing primes, this produces the sequence OEIS A217255 <http://oeis.org/A217255>.
is_extra_strong_lucas_pseudoprime
Takes a positive number as input, and returns 1 if the input passes the extra strong Lucas test (as
defined in Grantham 2000 <http://www.ams.org/mathscinet-getitem?mr=1680879>). This test has more
stringent conditions than the strong Lucas test, and produces about 60% fewer pseudoprimes. Performance
is typically 20-30% faster than the strong Lucas test.
The parameters are selected using the Baillie-OEIS method <http://oeis.org/A217719> method: increment "P"
from 3 until "jacobi(D,n) = -1". Removing primes, this produces the sequence OEIS A217719
<http://oeis.org/A217719>.
is_almost_extra_strong_lucas_pseudoprime
This is similar to the "is_extra_strong_lucas_pseudoprime" function, but does not calculate "U", so is a
little faster, but also weaker. 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 (an integer between 1 and 256) indicates the increment amount for "P"
parameter selection. The default value of 1 yields the parameter selection described in
"is_extra_strong_lucas_pseudoprime", creating a pseudoprime sequence which is a superset of the latter's
pseudoprime sequence OEIS A217719 <http://oeis.org/A217719>. A value of 2 yields the method used by 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_frobenius_underwood_pseudoprime
Takes a positive number as input, and returns 1 if the input passes the minimal lambda+2 test (see
Underwood 2012 "Quadratic Compositeness Tests"), where "(L+2)^(n-1) = 5 + 2x mod (n, L^2 - Lx + 1)". The
computational cost for this is between the cost of 2 and 3 strong pseudoprime tests. There are no known
counterexamples, but this is not a well studied test.
miller_rabin_random
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 should not be used in place of "is_prob_prime", "is_prime", or "is_provable_prime". Those functions
will be faster and provide better results than running "k" Miller-Rabin tests. This function can be used
if one wants more assurances for non-proven primes, such as for cryptographic uses where the size is
large enough that proven primes are not desired.
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 64-bit input (native or bignum), this uses either a deterministic set of Miller-Rabin tests (1, 2, or
3 tests) or a strong BPSW test consisting of a single base-2 strong probable prime test followed by a
strong Lucas test. This has been verified with Jan Feitsma's 2-PSP database to produce no false results
for 64-bit inputs. Hence the result will always be 0 (composite) or 2 (prime).
For inputs larger than "2^64", an extra-strong Baillie-PSW primality test is performed (also called BPSW
or BSW). This is a probabilistic test, so only 0 (composite) and 1 (probably prime) are returned. 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.
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). Normally one of the "is_prime" in Math::Prime::Util or
"is_prob_prime" in Math::Prime::Util functions will suffice, but those functions do pre-tests to find
easy composites. If you know this is not necessary, then calling "is_bpsw_prime" may save a small amount
of time.
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). This gives it the same return values as "is_prime" and "is_prob_prime". Note that
numbers below 2^64 are considered proven by the deterministic set of Miller-Rabin bases or the BPSW test.
Both of these have been tested for all small (64-bit) composites and do not return false positives.
Using the Math::Prime::Util::GMP module is highly recommended for doing primality proofs, as it is much,
much faster. The pure Perl code is just not fast for this type of operation, nor does it have the best
algorithms. It should suffice for proofs of up to 40 digit primes, while the latest MPU::GMP works for
primes of hundreds of digits (thousands with an optional larger polynomial set).
The pure Perl implementation uses theorem 5 of BLS75 (Brillhart, Lehmer, and Selfridge's 1975 paper), an
improvement on the Pocklington-Lehmer test. This requires "n-1" to be factored to "(n/2)^(1/3))". This
is often fast, but as "n" gets larger, it takes exponentially longer to find factors.
Math::Prime::Util::GMP implements both the BLS75 theorem 5 test as well as ECPP (elliptic curve primality
proving). It will typically try a quick "n-1" proof before using ECPP. Certificates are available with
either method. This results in proofs of 200-digit primes in under 1 second on average, and many
hundreds of digits are possible. This makes it significantly faster than Pari 2.1.7's "is_prime(n,1)"
which is the default for Math::Pari.
prime_certificate
my $cert = prime_certificate($n);
say verify_prime($cert) ? "proven prime" : "not prime";
Given a positive integer "n" as input, returns a primality certificate as a multi-line string. If we
could not prove "n" prime, an empty string is returned ("n" may or may not be composite). This may be
examined or given to "verify_prime" for verification. The latter function contains the description of
the format.
is_provable_prime_with_cert
Given a positive integer as input, returns a two element array containing the result of
"is_provable_prime":
0 definitely composite
1 probably prime
2 definitely prime and a primality certificate like "prime_certificate". The certificate will be an
empty string if the first element is not 2.
verify_prime
my $cert = prime_certificate($n);
say verify_prime($cert) ? "proven prime" : "not prime";
Given a primality certificate, returns either 0 (not verified) or 1 (verified). Most computations are
done using pure Perl with Math::BigInt, so you probably want to install and use Math::BigInt::GMP, and
ECPP certificates will be faster with Math::Prime::Util::GMP for its elliptic curve computations.
If the certificate is malformed, the routine will carp a warning in addition to returning 0. If the
"verbose" option is set (see "prime_set_config") then if the validation fails, the reason for the failure
is printed in addition to returning 0. If the "verbose" option is set to 2 or higher, then a message
indicating success and the certificate type is also printed.
A certificate may have arbitrary text before the beginning (the primality routines from this module will
not have any extra text, but this way verbose output from the prover can be safely stored in a
certificate). The certificate begins with the line:
[MPU - Primality Certificate]
All lines in the certificate beginning with "#" are treated as comments and ignored, as are blank lines.
A version number may follow, such as:
Version 1.0
For all inputs, base 10 is the default, but at any point this may be changed with a line like:
Base 16
where allowed bases are 10, 16, and 62. This module will only use base 10, so its routines will not
output Base commands.
Next, we look for (using "100003" as an example):
Proof for:
N 100003
where the text "Proof for:" indicates we will read an "N" value. Skipping comments and blank lines, the
next line should be "N " followed by the number.
After this, we read one or more blocks. Each block is a proof of the form:
If Q is prime, then N is prime.
Some of the blocks have more than one Q value associated with them, but most only have one. Each block
has its own set of conditions which must be verified, and this can be done completely self-contained.
That is, each block is independent of the other blocks and may be processed in any order. To be a
complete proof, each block must successfully verify. The block types and their conditions are shown
below.
Finally, when all blocks have been read and verified, we must ensure we can construct a proof tree from
the set of blocks. The root of the tree is the initial "N", and for each node (block), all "Q" values
must either have a block using that value as its "N" or "Q" must be less than "2^64" and pass BPSW.
Some other certificate formats (e.g. Primo) use an ordered chain, where the first block must be for the
initial "N", a single "Q" is given which is the implied "N" for the next block, and so on. This
simplifies validation implementation somewhat, and removes some redundant information from the
certificate, but has no obvious way to add proof types such as Lucas or the various BLS75 theorems that
use multiple factors. I decided that the most general solution was to have the certificate contain the
set in any order, and let the verifier do the work of constructing the tree.
The blocks begin with the text "Type ..." where ... is the type. One or more values follow. The defined
types are:
"Small"
Type Small
N 5791
N must be less than 2^64 and be prime (use BPSW or deterministic M-R).
"BLS3"
Type BLS3
N 2297612322987260054928384863
Q 16501461106821092981
A 5
A simple n-1 style proof using BLS75 theorem 3. This block verifies if:
a Q is odd
b Q > 2
c Q divides N-1
. Let M = (N-1)/Q
d MQ+1 = N
e M > 0
f 2Q+1 > sqrt(N)
g A^((N-1)/2) mod N = N-1
h A^(M/2) mod N != N-1
"Pocklington"
Type Pocklington
N 2297612322987260054928384863
Q 16501461106821092981
A 5
A simple n-1 style proof using generalized Pocklington. This is more restrictive than BLS3 and much
more than BLS5. This is Primo's type 1, and this module does not currently generate these blocks.
This block verifies if:
a Q divides N-1
. Let M = (N-1)/Q
b M > 0
c M < Q
d MQ+1 = N
e A > 1
f A^(N-1) mod N = 1
g gcd(A^M - 1, N) = 1
"BLS15"
Type BLS15
N 8087094497428743437627091507362881
Q 175806402118016161687545467551367
LP 1
LQ 22
A simple n+1 style proof using BLS75 theorem 15. This block verifies if:
a Q is odd
b Q > 2
c Q divides N+1
. Let M = (N+1)/Q
d MQ-1 = N
e M > 0
f 2Q-1 > sqrt(N)
. Let D = LP*LP - 4*LQ
g D != 0
h Jacobi(D,N) = -1
. Note: V_{k} indicates the Lucas V sequence with LP,LQ
i V_{m/2} mod N != 0
j V_{(N+1)/2} mod N == 0
"BLS5"
Type BLS5
N 8087094497428743437627091507362881
Q[1] 98277749
Q[2] 3631
A[0] 11
----
A more sophisticated n-1 proof using BLS theorem 5. This requires N-1 to be factored only to
"(N/2)^(1/3)". While this looks much more complicated, it really isn't much more work. The biggest
drawback is just that we have multiple Q values to chain rather than a single one. This block
verifies if:
a N > 2
b N is odd
. Note: the block terminates on the first line starting with a C<->.
. Let Q[0] = 2
. Let A[i] = 2 if Q[i] exists and A[i] does not
c For each i (0 .. maxi):
c1 Q[i] > 1
c2 Q[i] < N-1
c3 A[i] > 1
c4 A[i] < N
c5 Q[i] divides N-1
. Let F = N-1 divided by each Q[i] as many times as evenly possible
. Let R = (N-1)/F
d F is even
e gcd(F, R) = 1
. Let s = integer part of R / 2F
. Let f = fractional part of R / 2F
. Let P = (F+1) * (2*F*F + (r-1)*F + 1)
f n < P
g s = 0 OR r^2-8s is not a perfect square
h For each i (0 .. maxi):
h1 A[i]^(N-1) mod N = 1
h2 gcd(A[i]^((N-1)/Q[i])-1, N) = 1
"ECPP"
Type ECPP
N 175806402118016161687545467551367
A 96642115784172626892568853507766
B 111378324928567743759166231879523
M 175806402118016177622955224562171
Q 2297612322987260054928384863
X 3273750212
Y 82061726986387565872737368000504
An elliptic curve primality block, typically generated with an Atkin/Morain ECPP implementation, but
this should be adequate for anything using the Atkin-Goldwasser-Kilian-Morain style certificates.
Some basic elliptic curve math is needed for these. This block verifies if:
. Note: A and B are allowed to be negative, with -1 not uncommon.
. Let A = A % N
. Let B = B % N
a N > 0
b gcd(N, 6) = 1
c gcd(4*A^3 + 27*B^2, N) = 1
d Y^2 mod N = X^3 + A*X + B mod N
e M >= N - 2*sqrt(N) + 1
f M <= N + 2*sqrt(N) + 1
g Q > (N^(1/4)+1)^2
h Q < N
i M != Q
j Q divides M
. Note: EC(A,B,N,X,Y) is the point (X,Y) on Y^2 = X^3 + A*X + B, mod N
. All values work in affine coordinates, but in theory other
. representations work just as well.
. Let POINT1 = (M/Q) * EC(A,B,N,X,Y)
. Let POINT2 = M * EC(A,B,N,X,Y) [ = Q * POINT1 ]
k POINT1 is not the identity
l POINT2 is the identity
is_aks_prime
say "$n is definitely prime" if is_aks_prime($n);
Takes a positive number as input, and returns 1 if the input passes the Agrawal-Kayal-Saxena (AKS)
primality test. This is a deterministic unconditional primality test which runs in polynomial time for
general input.
While this is an important theoretical algorithm, and makes an interesting example, it is hard to
overstate just how impractically slow it is in practice. It is not used for any purpose in non-
theoretical work, as it is literally millions of times slower than other algorithms. From R.P. Brent,
2010: "AKS is not a practical algorithm. ECPP is much faster." We have ECPP, and indeed it is much
faster.
lucas_sequence
my($U, $V, $Qk) = lucas_sequence($n, $P, $Q, $k)
Computes "U_k", "V_k", and "Q_k" for the Lucas sequence defined by "P","Q", modulo "n". The modular
Lucas sequence is used in a number of primality tests and proofs. The following conditions must hold: "
D = P*P - 4*Q != 0" ; " 0 < P < n" ; " Q < n" ; " k >= 0" ; " n >= 2".
gcd
Given a list of integers, returns the greatest common divisor. This is often used to test for
coprimality <https://oeis.org/wiki/Coprimality>.
lcm
Given a list of integers, returns the least common multiple. Note that we follow the semantics of
Mathematica, Pari, and Perl 6, re:
lcm(0, n) = 0 Any zero in list results in zero return
lcm(n,-m) = lcm(n, m) We use the absolute values
moebius
say "$n is square free" if moebius($n) != 0;
$sum += moebius($_) for (1..200); say "Mertens(200) = $sum";
Returns X(n), the Moebius 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 Moebius function for every n from low to high inclusive. Large values of high will
result in a lot of memory use. The algorithm used for ranges is Deleglise and Rivat (1996) algorithm
4.1, which is a segmented version of Lioen and van de Lune (1994) algorithm 3.2.
The return values are read-only constants. This should almost never come up, but it means trying to
modify aliased return values will cause an exception (modifying the returned scalar or array is fine).
mertens
say "Mertens(10M) = ", mertens(10_000_000); # = 1037
Returns M(n), the Mertens function for a non-negative integer input. This function is defined as
"sum(moebius(1..n))", but calculated more efficiently for large inputs. For example, computing
Mertens(100M) takes:
time approx mem
0.3s 0.1MB mertens(100_000_000)
1.2s 890MB List::Util::sum(moebius(1,100_000_000))
77s 0MB $sum += moebius($_) for 1..100_000_000
The summation of individual terms via factoring is quite expensive in time, though uses O(1) space.
Using the range version of moebius is much faster, but returns a 100M element array which is not good for
memory with this many items. In comparison, this function will generate the equivalent output via a
sieving method that is relatively sparse memory and very fast. The current method is a simple "n^1/2"
version of Deleglise and Rivat (1996), which involves calculating all moebius values to "n^1/2", which in
turn will require prime sieving to "n^1/4".
Various algorithms exist for this, using differing quantities of X(n). The simplest way is to
efficiently sum all "n" values. Benito and Varona (2008) show a clever and simple method that only
requires "n/3" values. Deleglise and Rivat (1996) describe a segmented method using only "n^1/3" values.
The current implementation does a simple non-segmented "n^1/2" version of their method. Kuznetsov (2011)
gives an alternate method that he indicates is even faster. Lastly, one of the advanced prime count
algorithms could be theoretically used to create a faster solution.
euler_phi
say "The Euler totient of $n is ", euler_phi($n);
Returns X(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, "euler_phi" will return 0 for all "n < 1".
This follows the logic used by SAGE. Mathematica and Pari return "euler_phi(-n)" for "n < 0".
Mathematica returns 0 for "n = 0" while Pari raises an exception.
If called with two arguments, they define a range "low" to "high", and the function returns an array with
the totient of every n from low to high inclusive.
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($n)" This counts the number of k-tuples less than or equal to n
that form a coprime tuple with n. As with "euler_phi", 0 is returned for all "n < 1". This function can
be used to generate some other useful functions, such as the Dedikind psi function, where "psi(n) =
J(2,n) / J(1,n)".
exp_mangoldt
say "exp(lambda($_)) = ", exp_mangoldt($_) for 1 .. 100;
Returns EXP(X(n)), the exponential of the Mangoldt function (also known as von Mangoldt's function) for
an integer value. The Mangoldt function is equal to log p if n is prime or a power of a prime, and 0
otherwise. We return the exponential so all results are integers. Hence the return value for
"exp_mangoldt" is:
p if n = p^m for some prime p and integer m >= 1
1 otherwise.
liouville
Returns X(n), the Liouville function for a non-negative integer input. This is -1 raised to X(n) (the
total number of prime factors).
chebyshev_theta
say chebyshev_theta(10000);
Returns X(n), the first Chebyshev function for a non-negative integer input. This is the sum of the
logarithm of each prime where "p <= n". An alternate computation is as the logarithm of n primorial.
Hence these functions:
use List::Util qw/sum/; use Math::BigFloat;
sub c1a { 0+sum( map { log($_) } @{primes(shift)} ) }
sub c1b { Math::BigFloat->new(primorial(shift))->blog }
yield similar results, albeit slower and using more memory.
chebyshev_psi
say chebyshev_psi(10000);
Returns X(n), the second Chebyshev function for a non-negative integer input. This is the sum of the
logarithm of each prime power where "p^k <= n" for an integer k. An alternate computation is as the
summatory Mangoldt function. Another alternate computation is as the logarithm of LCM(1,2,...,n). Hence
these functions:
use List::Util qw/sum/; use Math::BigFloat;
sub c2a { 0+sum( map { log(exp_mangoldt($_)) } 1 .. shift ) }
sub c2b { Math::BigFloat->new(consecutive_integer_lcm(shift))->blog }
yield similar results, albeit slower and using more memory.
divisor_sum
say "Sum of divisors of $n:", divisor_sum( $n );
say "sigma_2($n) = ", divisor_sum($n, 2);
say "Number of divisors: sigma_0($n) = ", divisor_sum($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.
The second argument may also be a code reference, which is called for each divisor and the results are
summed. This allows computation of other functions, but will be less efficient than using the numeric
second argument. This corresponds to Pari/GP's "sumdiv" function.
An example of the 5th Jordan totient (OEIS A059378):
divisor_sum( $n, sub { my $d=shift; $d**5 * moebius($n/$d); } );
though we have a function "jordan_totient" which is more efficient.
For numeric second arguments (sigma computations), the result will be a bigint if necessary. For the
code reference case, the user must take care to return bigints if overflow will be a concern.
primorial
$prim = primorial(11); # 11# = 2*3*5*7*11 = 2310
Returns the primorial "n#" of the positive integer input, defined as the product of the prime numbers
less than or equal to "n". This is the OEIS series A034386 <http://oeis.org/A034386>: primorial numbers
second definition.
primorial(0) == 1
primorial($n) == pn_primorial( prime_count($n) )
The result will be a Math::BigInt object if it is larger than the native bit size.
Be careful about which version ("primorial" or "pn_primorial") matches the definition you want to use.
Not all sources agree on the terminology, though they should give a clear definition of which of the two
versions they mean. OEIS, Wikipedia, and Mathworld are all consistent, and these functions should match
that terminology. This function should return the same result as the "mpz_primorial_ui" function added
in GMP 5.1.
pn_primorial
$prim = pn_primorial(5); # p_5# = 2*3*5*7*11 = 2310
Returns the primorial number "p_n#" of the positive integer input, defined as the product of the first
"n" prime numbers (compare to the factorial, which is the product of the first "n" natural numbers).
This is the OEIS series A002110 <http://oeis.org/A002110>: primorial numbers first definition.
pn_primorial(0) == 1
pn_primorial($n) == primorial( nth_prime($n) )
The result will be a Math::BigInt object if it is larger than the native bit size.
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 a factorial.
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:
65 Integer::Partition
10_000 MPU pure Perl partitions
200_000 MPU GMP partitions
22_000_000 Pari's numbpart
500_000_000 Jonathan Bober's partitions_c.cc v0.6
If you want the enumerated partitions, see Integer::Partition. It uses a memory efficient iterator and
is very fast for enumeration. It is not practical for producing large partition numbers as seen above.
carmichael_lambda
Returns the Carmichael function (also called the reduced totient function, or Carmichael X(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>.
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.
znorder
$order = znorder(2, next_prime(10**19)-6);
Given two positive integers "a" and "n", returns the multiplicative order of "a" modulo "n". This is the
smallest positive integer "k" such that "a^k X 1 mod n". Returns 1 if "a = 1". Returns undef if "a = 0"
or if "a" and "n" are not coprime, since no value will result in 1 mod n. This corresponds to Pari's
"znorder(Mod(a,n))" function and Mathematica's "MultiplicativeOrder[n]" function.
znprimroot
Given a positive integer "n", returns the smallest primitive root of "(Z/nZ)^*", or "undef" if no root
exists. A root exists when "euler_phi($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.
znlog
$k = znlog($a, $g, $p)
Returns the integer "k" that solves the equation "a = g^k mod p", or undef if no solution is found. This
is the discrete logarithm problem. The implementation in this version is not very useful, but may be
improved.
legendre_phi
$phi = legendre_phi(1000000000, 41);
Given a non-negative integer "n" and a non-negative prime number "a", returns the Legendre phi function
(also called Legendre's sum). This is the count of positive integers <= "n" which are not divisible by
any of the first "a" primes.
RANDOM PRIMES
random_prime
my $small_prime = random_prime(1000); # random prime <= limit
my $rand_prime = random_prime(100, 10000); # random prime within a range
Returns a pseudo-randomly selected prime that will be greater than or equal to the lower limit and less
than or equal to the upper limit. If no lower limit is given, 2 is implied. Returns undef if no primes
exist within the range.
The goal is to return a uniform distribution of the primes in the range, meaning for each prime in the
range, the chances are equally likely that it will be seen. This is removes from consideration such
algorithms as "PRIMEINC", which although efficient, gives very non-random output. This also implies that
the numbers will not be evenly distributed, since the primes are not evenly distributed. Stated
differently, the random prime functions return a uniformly selected prime from the set of primes within
the range. Hence given "random_prime(1000)", the numbers 2, 3, 487, 631, and 997 all have the same
probability of being returned.
For small numbers, a random index selection is done, which gives ideal uniformity and is very efficient
with small inputs. For ranges larger than this ~16-bit threshold but within the native bit size, a Monte
Carlo method is used (multiple calls to "irand" will be made if necessary). This also gives ideal
uniformity and can be very fast for reasonably sized ranges. For even larger numbers, we partition the
range, choose a random partition, then select a random prime from the partition. This gives some loss of
uniformity but results in many fewer bits of randomness being consumed as well as being much faster.
If an "irand" function has been set via "prime_set_config", it will be used to construct any ranged
random numbers needed. The function should return a uniformly random 32-bit integer, which is how the
irand functions exported by Math::Random::Secure, Math::Random::MT, Math::Random::ISAAC, and most other
modules behave.
If no "irand" function was set, then Bytes::Random::Secure is used with a non-blocking seed. This will
create good quality random numbers, so there should be little reason to change unless one is generating
long-term keys, where using the blocking random source may be preferred.
Examples of various ways to set your own irand function:
# System rand. You probably don't want to do this.
prime_set_config(irand => sub { int(rand(4294967296)) });
# Math::Random::Secure. Uses ISAAC and strong seed methods.
use Math::Random::Secure;
prime_set_config(irand => \&Math::Random::Secure::irand);
# Bytes::Random::Secure (OO interface with full control of options):
use Bytes::Random::Secure ();
BEGIN {
my $rng = Bytes::Random::Secure->new( Bits => 512 );
sub irand { return $rng->irand; }
}
prime_set_config(irand => \&irand);
# Crypt::Random. Uses Pari and /dev/random. Very slow.
use Crypt::Random qw/makerandom/;
prime_set_config(irand => sub { makerandom(Size=>32, Uniform=>1); });
# Mersenne Twister. Very fast, decent RNG, auto seeding.
use Math::Random::MT::Auto;
prime_set_config(irand=>sub {Math::Random::MT::Auto::irand() & 0xFFFFFFFF});
# Go back to MPU's default configuration
prime_set_config(irand => undef);
random_ndigit_prime
say "My 4-digit prime number is: ", random_ndigit_prime(4);
Selects a random n-digit prime, where the input is an integer number of digits. One of the primes within
that range (e.g. 1000 - 9999 for 4-digits) will be uniformly selected using the "irand" function as
described above.
If the number of digits is greater than or equal to the maximum native type, then the result will be
returned as a BigInt. However, if the "nobigint" configuration option is on, then output will be
restricted to native size numbers, and requests for more digits than natively supported will result in an
error. For better performance with large bit sizes, install Math::Prime::Util::GMP.
random_nbit_prime
my $bigprime = random_nbit_prime(512);
Selects a random n-bit prime, where the input is an integer number of bits. A prime with the nth bit set
will be uniformly selected, with randomness supplied via calls to the "irand" function as described
above.
For bit sizes of 64 and lower, "random_prime" is used, which gives completely uniform results in this
range. For sizes larger than 64, Algorithm 1 of Fouque and Tibouchi (2011) is used, wherein we select a
random odd number for the lower bits, then loop selecting random upper bits until the result is prime.
This allows a more uniform distribution than the general "random_prime" case while running slightly
faster (in contrast, for large bit sizes "random_prime" selects a random upper partition then loops on
the values within the partition, which very slightly skews the results towards smaller numbers).
The "irand" function is used for randomness, so all the discussion in "random_prime" about that applies
here. The result will be a BigInt if the number of bits is greater than the native bit size. For better
performance with large bit sizes, install Math::Prime::Util::GMP.
random_strong_prime
my $bigprime = random_strong_prime(512);
Constructs an n-bit strong prime using Gordon's algorithm. We consider a strong prime p to be one where
• p is large. This function requires at least 128 bits.
• p-1 has a large prime factor r.
• p+1 has a large prime factor s
• r-1 has a large prime factor t
Using a strong prime in cryptography guards against easy factoring with algorithms like Pollard's Rho.
Rivest and Silverman (1999) present a case that using strong primes is unnecessary, and most modern
cryptographic systems agree. First, the smoothness does not affect more modern factoring methods such as
ECM. Second, modern factoring methods like GNFS are far faster than either method so make the point
moot. Third, due to key size growth and advances in factoring and attacks, for practical purposes, using
large random primes offer security equivalent to strong primes.
Similar to "random_nbit_prime", the result will be a BigInt if the number of bits is greater than the
native bit size. For better performance with large bit sizes, install Math::Prime::Util::GMP.
random_proven_prime
my $bigprime = random_proven_prime(512);
Constructs an n-bit random proven prime. Internally this may use
"is_provable_prime"("random_nbit_prime") or "random_maurer_prime" depending on the platform and bit size.
random_proven_prime_with_cert
my($n, $cert) = random_proven_prime_with_cert(512)
Similar to "random_proven_prime", but returns a two-element array containing the n-bit provable prime
along with a primality certificate. The certificate is the same as produced by "prime_certificate" or
"is_provable_prime_with_cert", and can be parsed by "verify_prime" or any other software that understands
MPU primality certificates.
random_maurer_prime
my $bigprime = random_maurer_prime(512);
Construct an n-bit provable prime, using the FastPrime algorithm of Ueli Maurer (1995). This is the same
algorithm used by Crypt::Primes. Similar to "random_nbit_prime", the result will be a BigInt if the
number of bits is greater than the native bit size. For better performance with large bit sizes, install
Math::Prime::Util::GMP.
The differences between this function and that in Crypt::Primes are described in the "SEE ALSO" section.
Internally this additionally runs the BPSW probable prime test on every partial result, and constructs a
primality certificate for the final result, which is verified. These provide additional checks that the
resulting value has been properly constructed.
An alternative to this function is to run "is_provable_prime" on the result of "random_nbit_prime", which
will provide more diversity and will be faster up to 512 or so bits. Maurer's method should be much
faster for large bit sizes (larger than 2048). If you don't need absolutely proven results, then using
"random_nbit_prime" followed by additional tests ("is_strong_pseudoprime" and/or
"is_frobenius_underwood_pseudoprime") should be much faster.
random_maurer_prime_with_cert
my($n, $cert) = random_maurer_prime_with_cert(512)
As with "random_maurer_prime", but returns a two-element array containing the n-bit provable prime along
with a primality certificate. The certificate is the same as produced by "prime_certificate" or
"is_provable_prime_with_cert", and can be parsed by "verify_prime" or any other software that understands
MPU primality certificates. The proof construction consists of a single chain of "BLS3" types.
UTILITY FUNCTIONS
prime_precalc
prime_precalc( 1_000_000_000 );
Let the module prepare for fast operation up to a specific number. It is not necessary to call this, but
it gives you more control over when memory is allocated and gives faster results for multiple calls in
some cases. In the current implementation this will calculate a sieve for all numbers up to the
specified number.
prime_memfree
prime_memfree;
Frees any extra memory the module may have allocated. Like with "prime_precalc", it is not necessary to
call this, but if you're done making calls, or want things cleanup up, you can use this. The object
method might be a better choice for complicated uses.
Math::Prime::Util::MemFree->new
my $mf = Math::Prime::Util::MemFree->new;
# perform operations. When $mf goes out of scope, memory will be recovered.
This is a more robust way of making sure any cached memory is freed, as it will be handled by the last
"MemFree" object leaving scope. This means if your routines were inside an eval that died, things will
still get cleaned up. If you call another function that uses a MemFree object, the cache will stay in
place because you still have an object.
prime_get_config
my $cached_up_to = prime_get_config->{'precalc_to'};
Returns a reference to a hash of the current settings. The hash is copy of the configuration, so
changing it has no effect. The settings include:
precalc_to primes up to this number are calculated
maxbits the maximum number of bits for native operations
xs 0 or 1, indicating the XS code is available
gmp 0 or 1, indicating GMP code is available
maxparam the largest value for most functions, without bigint
maxdigits the max digits in a number, without bigint
maxprime the largest representable prime, without bigint
maxprimeidx the index of maxprime, without bigint
assume_rh whether to assume the Riemann hypothesis (default 0)
prime_set_config
prime_set_config( assume_rh => 1 );
Allows setting of some parameters. Currently the only parameters are:
xs Allows turning off the XS code, forcing the Pure Perl
code to be used. Set to 0 to disable XS, set to 1 to
re-enable. You probably will never want to do this.
gmp Allows turning off the use of L<Math::Prime::Util::GMP>,
which means using Pure Perl code for big numbers. Set
to 0 to disable GMP, set to 1 to re-enable.
You probably will never want to do this.
assume_rh Allows functions to assume the Riemann hypothesis is
true if set to 1. This defaults to 0. Currently this
setting only impacts prime count lower and upper
bounds, but could later be applied to other areas such
as primality testing. A later version may also have a
way to indicate whether no RH, RH, GRH, or ERH is to
be assumed.
irand Takes a code ref to an irand function returning a
uniform number between 0 and 2**32-1. This will be
used for all random number generation in the module.
FACTORING FUNCTIONS
factor
my @factors = factor(3_369_738_766_071_892_021);
# returns (204518747,16476429743)
Produces the prime factors of a positive number input, in numerical order. The product of the returned
factors will be equal to the input. "n = 1" will return an empty list, and "n = 0" will return 0. This
matches Pari.
In scalar context, returns X(n), the total number of prime factors (OEIS A001222
<http://oeis.org/A001222>). This corresponds to Pari's bigomega(n) function and Mathematica's
"PrimeOmega[n]" function. This is same result that we would get if we evaluated the resulting array in
scalar context.
The current algorithm for non-bigints is a sequence of small trial division, a few rounds of Pollard's
Rho, SQUFOF, Pollard's p-1, Hart's OLF, a long run of Pollard's Rho, and finally trial division if
anything survives. This process is repeated for each non-prime factor. In practice, it is very rare to
require more than the first Rho + SQUFOF to find a factor, and I have not seen anything go to the last
step.
Factoring bigints works with pure Perl, and can be very handy on 32-bit machines for numbers just over
the 32-bit limit, but it can be very slow for "hard" numbers. Installing the Math::Prime::Util::GMP
module will speed up bigint factoring a lot, and all future effort on large number factoring will be in
that module. If you do not have that module for some reason, use the GMP or Pari version of bigint if
possible (e.g. "use bigint try => 'GMP,Pari'"), which will run 2-3x faster (though still 100x slower than
the real GMP code).
factor_exp
my @factor_exponent_pairs = factor_exp(29513484000);
# returns ([2,5], [3,4], [5,3], [7,2], [11,1], [13,2])
# factor(29513484000)
# returns (2,2,2,2,2,3,3,3,3,5,5,5,7,7,11,13,13)
Produces pairs of prime factors and exponents in numerical factor order. This is more convenient for
some algorithms. This is the same form that Mathematica's "FactorInteger[n]" and Pari/GP's "factorint"
functions return. Note that Math::Pari transposes the Pari result matrix.
In scalar context, returns X(n), the number of unique prime factors (OEIS A001221
<http://oeis.org/A001221>). This corresponds to Pari's omega(n) function and Mathematica's "PrimeNu[n]"
function. This is same result that we would get if we evaluated the resulting array in scalar context.
The internals are identical to "factor", so all comments there apply. Just the way the factors are
arranged is different.
divisors
all_factors
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 uniqued sorted list. The result is
identical to that of Pari's "divisors" and Mathematica's "Divisors[n]" functions.
In scalar context this returns the sigma0 function, the sigma function (see Hardy and Wright section
16.7, or OEIS A000203). This is the same result as evaluating the array in scalar context.
Also see the "for_divisors" functions for looping over the divisors.
"all_factors" is the deprecated name for this function.
trial_factor
my @factors = trial_factor($n);
Produces the prime factors of a positive number input. The factors will be in numerical order. For
large inputs this will be very slow.
fermat_factor
my @factors = fermat_factor($n);
Produces factors, not necessarily prime, of the positive number input. The particular algorithm is
Knuth's algorithm C. For small inputs this will be very fast, but it slows down quite rapidly as the
number of digits increases. It is very fast for inputs with a factor close to the midpoint (e.g. a
semiprime p*q where p and q are the same number of digits).
holf_factor
my @factors = holf_factor($n);
Produces factors, not necessarily prime, of the positive number input. An optional number of rounds can
be given as a second parameter. It is possible the function will be unable to find a factor, in which
case a single element, the input, is returned. This uses Hart's One Line Factorization with no
premultiplier. It is an interesting alternative to Fermat's algorithm, and there are some inputs it can
rapidly factor. In the long run it has the same advantages and disadvantages as Fermat's method.
squfof_factor
my @factors = squfof_factor($n);
Produces factors, not necessarily prime, of the positive number input. An optional number of rounds can
be given as a second parameter. It is possible the function will be unable to find a factor, in which
case a single element, the input, is returned. This function typically runs very fast.
prho_factor
pbrent_factor
my @factors = prho_factor($n);
my @factors = pbrent_factor($n);
# Use a very small number of rounds
my @factors = prho_factor($n, 1000);
Produces factors, not necessarily prime, of the positive number input. An optional number of rounds can
be given as a second parameter. These attempt to find a single factor using Pollard's Rho algorithm,
either the original version or Brent's modified version. These are more specialized algorithms usually
used for pre-factoring very large inputs, as they are very fast at finding small factors.
pminus1_factor
my @factors = pminus1_factor($n);
my @factors = pminus1_factor($n, 1_000); # set B1 smoothness
my @factors = pminus1_factor($n, 1_000, 50_000); # set B1 and B2
Produces factors, not necessarily prime, of the positive number input. This is Pollard's "p-1" method,
using two stages with default smoothness settings of 1_000_000 for B1, and "10 * B1" for B2. This method
can rapidly find a factor "p" of "n" where "p-1" is smooth (it has no large factors).
pplus1_factor
my @factors = pplus1_factor($n);
my @factors = pplus1_factor($n, 1_000); # set B1 smoothness
Produces factors, not necessarily prime, of the positive number input. This is Williams' "p+1" method,
using one stage and two predefined initial points.
MATHEMATICAL FUNCTIONS
ExponentialIntegral
my $Ei = ExponentialIntegral($x);
Given a non-zero floating point input "x", this returns the real-valued exponential integral of "x",
defined as the integral of "e^t/t dt" from "-infinity" to "x".
If the bignum module has been loaded, all inputs will be treated as if they were Math::BigFloat objects.
For non-BigInt/BigFloat objects, the result should be accurate to at least 14 digits.
For BigInt / BigFloat objects, we first check to see if Math::MPFR is available. If so, then it is used
since it is very fast and has high accuracy. Accuracy when using MPFR will be equal to the "accuracy()"
value of the input (or the default BigFloat accuracy, which is 40 by default).
MPFR is used for positive inputs only. If Math::MPFR is not available or the input is negative, then
other methods are used: continued fractions ("x < -1"), rational Chebyshev approximation (" -1 < x < 0"),
a convergent series (small positive "x"), or an asymptotic divergent series (large positive "x").
Accuracy should be at least 14 digits.
LogarithmicIntegral
my $li = LogarithmicIntegral($x)
Given a positive floating point input, returns the floating point logarithmic integral of "x", defined as
the integral of "dt/ln t" from 0 to "x". If given a negative input, the function will croak. The
function returns 0 at "x = 0", and "-infinity" at "x = 1".
This is often known as li(x). A related function is the offset logarithmic integral, sometimes known as
Li(x) which avoids the singularity at 1. It may be defined as "Li(x) = li(x) - li(2)". Crandall and
Pomerance use the term "li0" for this function, and define "li(x) = Li0(x) - li0(2)". Due to this
terminology confusion, it is important to check which exact definition is being used.
If the bignum module has been loaded, all inputs will be treated as if they were Math::BigFloat objects.
For non-BigInt/BigFloat objects, the result should be accurate to at least 14 digits.
For BigInt / BigFloat objects, we first check to see if Math::MPFR is available. If so, then it is used,
as it will return results much faster and can be more accurate. Accuracy when using MPFR will be equal
to the "accuracy()" value of the input (or the default BigFloat accuracy, which is 40 by default).
MPFR is used for inputs greater than 1 only. If Math::MPFR is not installed or the input is less than 1,
results will be calculated as "Ei(ln x)".
RiemannZeta
my $z = RiemannZeta($s);
Given a floating point input "s" where "s >= 0", returns the floating point value of X(s)-1, where X(s)
is the Riemann zeta function. One is subtracted to ensure maximum precision for large values of "s".
The zeta function is the sum from k=1 to infinity of "1 / k^s". This function only uses real arguments,
so is basically the Euler Zeta function.
If the bignum module has been loaded, all inputs will be treated as if they were Math::BigFloat objects.
For non-BigInt/BigFloat objects, the result should be accurate to at least 14 digits. The XS code uses a
rational Chebyshev approximation between 0.5 and 5, and a series for other values. The PP code uses an
identical series for all values.
For BigInt / BigFloat objects, we first check to see if the Math::MPFR module is installed. If so, then
it is used, as it will return results much faster and can be more accurate. Accuracy when using MPFR
will be equal to the "accuracy()" value of the input (or the default BigFloat accuracy, which is 40 by
default).
If Math::MPFR is not installed, then results are calculated using either Borwein (1991) algorithm 2, or
the basic series. Full input accuracy is attempted, but Math::BigFloat RT 43692
<https://rt.cpan.org/Ticket/Display.html?id=43692> produces incorrect high-accuracy computations without
the fix. It is also very slow. I highly recommend installing Math::MPFR for BigFloat computations.
RiemannR
my $r = RiemannR($x);
Given a positive non-zero floating point input, returns the floating point value of Riemann's R function.
Riemann's R function gives a very close approximation to the prime counting function.
If the bignum module has been loaded, all inputs will be treated as if they were Math::BigFloat objects.
For non-BigInt/BigFloat objects, the result should be accurate to at least 14 digits.
For BigInt / BigFloat objects, we first check to see if the Math::MPFR module is installed. If so, then
it is used, as it will return results much faster and can be more accurate. Accuracy when using MPFR
will be equal to the "accuracy()" value of the input (or the default BigFloat accuracy, which is 40 by
default). Accuracy without MPFR should be 35 digits.
EXAMPLES
Print strong pseudoprimes to base 17 up to 10M:
# Similar to A001262's isStrongPsp function, but much faster
perl -MMath::Prime::Util=:all -E 'forcomposites { say if is_strong_pseudoprime($_,17) } 10000000;'
Print some primes above 64-bit range:
perl -MMath::Prime::Util=:all -Mbigint -E 'my $start=100000000000000000000; say join "\n", @{primes($start,$start+1000)}'
# Another way
perl -MMath::Prime::Util=:all -E 'forprimes { say } "100000000000000000039", "100000000000000000993"'
# Similar using Math::Pari:
# perl -MMath::Pari=:int,PARI,nextprime -E 'my $start = PARI "100000000000000000000"; my $end = $start+1000; my $p=nextprime($start); while ($p <= $end) { say $p; $p = nextprime($p+1); }'
Examining the X3(x) function of Planat and Sole (2011):
sub nu3 {
my $n = shift;
my $phix = chebyshev_psi($n);
my $nu3 = 0;
foreach my $nu (1..3) {
$nu3 += (moebius($nu)/$nu)*LogarithmicIntegral($phix**(1/$nu));
}
return $nu3;
}
say prime_count(1000000);
say prime_count_approx(1000000);
say nu3(1000000);
Construct and use a Sophie-Germain prime iterator:
sub make_sophie_germain_iterator {
my $p = shift || 2;
my $it = prime_iterator($p);
return sub {
do { $p = $it->() } while !is_prime(2*$p+1);
$p;
};
}
my $sgit = make_sophie_germain_iterator();
print $sgit->(), "\n" for 1 .. 10000;
Project Euler, problem 3 (Largest prime factor):
use Math::Prime::Util qw/factor/;
use bigint; # Only necessary for 32-bit machines.
say 0+(factor(600851475143))[-1]
Project Euler, problem 7 (10001st prime):
use Math::Prime::Util qw/nth_prime/;
say nth_prime(10_001);
Project Euler, problem 10 (summation of primes):
use Math::Prime::Util qw/forprimes/;
my $sum = 0;
forprimes { $sum += $_ } 2_000_000;
say $sum;
Project Euler, problem 21 (Amicable numbers):
use Math::Prime::Util qw/divisor_sum/;
sub dsum { my $n = shift; divisor_sum($n) - $n; }
my $sum = 0;
foreach my $a (1..10000) {
my $b = dsum($a);
$sum += $a + $b if $b > $a && dsum($b) == $a;
}
say $sum;
Project Euler, problem 41 (Pandigital prime), brute force command line:
perl -MMath::Prime::Util=primes -MList::Util=first -E 'say first { /1/&&/2/&&/3/&&/4/&&/5/&&/6/&&/7/} reverse @{primes(1000000,9999999)};'
Project Euler, problem 47 (Distinct primes factors):
use Math::Prime::Util qw/pn_primorial factor_exp/;
my $n = pn_primorial(4); # Start with the first 4-factor number
# factor_exp in scalar context returns the number of distinct prime factors
$n++ while (factor_exp($n) != 4 || factor_exp($n+1) != 4 || factor_exp($n+2) != 4 || factor_exp($n+3) != 4);
say $n;
Project Euler, problem 69, stupid brute force solution (about 1 second):
use Math::Prime::Util qw/euler_phi/;
my ($n, $max) = (0,0);
do {
my $ndivphi = $_ / euler_phi($_);
($n, $max) = ($_, $ndivphi) if $ndivphi > $max;
} for 1..1000000;
say "$n $max";
Here is the right way to do PE problem 69 (under 0.03s):
use Math::Prime::Util qw/pn_primorial/;
my $n = 0;
$n++ while pn_primorial($n+1) < 1000000;
say pn_primorial($n);
Project Euler, problem 187, stupid brute force solution, ~3 minutes:
use Math::Prime::Util qw/factor/;
my $nsemis = 0;
do { $nsemis++ if scalar factor($_) == 2; }
for 1 .. int(10**8)-1;
say $nsemis;
Here is the best way for PE187. Under 30 milliseconds from the command line:
use Math::Prime::Util qw/primes prime_count/;
use List::Util qw/sum/;
my $limit = shift || int(10**8);
my @primes = @{primes(int(sqrt($limit)))};
say sum( map { prime_count(int(($limit-1)/$primes[$_-1])) - $_ + 1 }
1 .. scalar @primes );
Produce the "matches" result from Math::Factor::XS without skipping:
use Math::Prime::Util qw/divisors/;
use Algorithm::Combinatorics qw/combinations_with_repetition/;
my $n = 139650;
my @matches = grep { $_->[0] * $_->[1] == $n && $_->[0] > 1 }
combinations_with_repetition( [divisors($n)], 2 );
Compute OEIS A054903 <http://oeis.org/A054903> just like CRG4's Pari example:
use Math::Prime::Util qw/forcomposite divisor_sum/;
forcomposites {
say if divisor_sum($_)+6 == divisor_sum($_+6)
} 9,1e7;
Construct the table shown in OEIS A046147 <http://oeis.org/A046147>:
use Math::Prime::Util qw/znorder euler_phi gcd/;
foreach my $n (1..100) {
if (!znprimroot($n)) {
say "$n -";
} else {
my $phi = euler_phi($n);
my @r = grep { gcd($_,$n) == 1 && znorder($_,$n) == $phi } 1..$n-1;
say "$n ", join(" ", @r);
}
}
PRIMALITY TESTING NOTES
Above "2^64", "is_prob_prime" performs an extra-strong BPSW test <http://en.wikipedia.org/wiki/Baillie- PSW_primality_test> which is fast (a little less than the time to perform 3 Miller-Rabin tests) and has no known counterexamples. If you trust the primality testing done by Pari, Maple, SAGE, FLINT, etc., then this function should be appropriate for you. "is_prime" will do the same BPSW test as well as some additional testing, making it slightly more time consuming but less likely to produce a false result. This is a little more stringent than Mathematica. "is_provable_prime" constructs a primality proof. If a certificate is requested, then either BLS75 theorem 5 or ECPP is performed. Without a certificate, the method is implementation specific (currently it is identical, but later releases may use APRCL). With Math::Prime::Util::GMP installed, this is quite fast through 300 or so digits. Math systems 30 years ago typically used Miller-Rabin tests with "k" bases (usually fixed bases, sometimes random) for primality testing, but these have generally been replaced by some form of BPSW as used in this module. See Pinch's 1993 paper for examples of why using "k" M-R tests leads to poor results. The three exceptions in common contemporary use I am aware of are: libtommath Uses the first "k" prime bases. This is problematic for cryptographic use, as there are known methods (e.g. Arnault 1994) for constructing counterexamples. The number of bases required to avoid false results is unreasonably high, hence performance is slow even if one ignores counterexamples. Unfortunately this is the multi-precision math library used for Perl 6 and at least one CPAN Crypto module. GMP/MPIR Uses a set of "k" static-random bases. The bases are randomly chosen using a PRNG that is seeded identically each call (the seed changes with each release). This offers a very slight advantage over using the first "k" prime bases, but not much. See, for example, Nicely's mpz_probab_prime_p pseudoprimes <http://www.trnicely.net/misc/mpzspsp.html> page. Math::Pari Pari 2.1.7 is the default version installed with the Math::Pari module. It uses 10 random M-R bases (the PRNG uses a fixed seed set at compile time). Pari 2.3.0 was released in May 2006 and it, like all later releases through at least 2.6.1, use BPSW / APRCL, after complaints of false results from using M-R tests. Basically the problem is that it is just too easy to get counterexamples from running "k" M-R tests, forcing one to use a very large number of tests (at least 20) to avoid frequent false results. Using the BPSW test results in no known counterexamples after 30+ years and runs much faster. It can be enhanced with one or more random bases if one desires, and will still be much faster. Using "k" fixed bases has another problem, which is that in any adversarial situation we can assume the inputs will be selected such that they are one of our counterexamples. Now we need absurdly large numbers of tests. This is like playing "pick my number" but the number is fixed forever at the start, the guesser gets to know everyone else's guesses and results, and can keep playing as long as they like. It's only valid if the players are completely oblivious to what is happening.
LIMITATIONS
Perl versions earlier than 5.8.0 have problems doing exact integer math. Some operations will flip
signs, and many operations will convert intermediate or output results to doubles, which loses precision
on 64-bit systems. This causes numerous functions to not work properly. The test suite will try to
determine if your Perl is broken (this only applies to really old versions of Perl compiled for 64-bit
when using numbers larger than "~ 2^49"). The best solution is updating to a more recent Perl.
The module is thread-safe and should allow good concurrency on all platforms that support Perl threads
except Win32. With Win32, either don't use threads or make sure "prime_precalc" is called before using
"primes", "prime_count", or "nth_prime" with large inputs. This is only an issue if you use non-Cygwin
Win32 and call these routines from within Perl threads.
SEE ALSO
This section describes other CPAN modules available that have some feature overlap with this one. Also
see the "REFERENCES" section. Please let me know if any of this information is inaccurate. Also note
that just because a module doesn't match what I believe are the best set of features, doesn't mean it
isn't perfect for someone else.
I will use SoE to indicate the Sieve of Eratosthenes, and MPU to denote this module (Math::Prime::Util).
Some quick alternatives I can recommend if you don't want to use MPU:
• Math::Prime::FastSieve is the alternative module I use for basic functionality with small integers.
It's fast and simple, and has a good set of features.
• Math::Primality is the alternative module I use for primality testing on bigints. The downside is
that it can be slow, and the functions other than primality tests are very slow.
• Math::Pari if you want the kitchen sink and can install it and handle using it. There are still some
functions it doesn't do well (e.g. prime count and nth_prime).
Math::Prime::XS has "is_prime" and "primes" functionality. There is no bigint support. The "is_prime"
function uses well-written trial division, meaning it is very fast for small numbers, but terribly slow
for large 64-bit numbers. MPU is similarly fast with small numbers, but becomes faster as the size
increases. MPXS's prime sieve is an unoptimized non-segmented SoE which returns an array. Sieve bases
larger than "10^7" start taking inordinately long and using a lot of memory (gigabytes beyond "10^10").
E.g. "primes(10**9, 10**9+1000)" takes 36 seconds with MPXS, but only 0.00015 seconds with MPU.
Math::Prime::FastSieve supports "primes", "is_prime", "next_prime", "prev_prime", "prime_count", and
"nth_prime". The caveat is that all functions only work within the sieved range, so are limited to about
"10^10". It uses a fast SoE to generate the main sieve. The sieve is 2-3x slower than the base sieve
for MPU, and is non-segmented so cannot be used for larger values. Since the functions work with the
sieve, they are very fast. The fast bit-vector-lookup functionality can be replicated in MPU using
"prime_precalc" but is not required.
Bit::Vector supports the "primes" and "prime_count" functionality in a somewhat similar way to
Math::Prime::FastSieve. It is the slowest of all the XS sieves, and has the most memory use. It is
faster than pure Perl code.
Crypt::Primes supports "random_maurer_prime" functionality. MPU has more options for random primes
(n-digit, n-bit, ranged, and strong) in addition to Maurer's algorithm. MPU does not have the critical
bug RT81858 <https://rt.cpan.org/Ticket/Display.html?id=81858>. MPU should have a more uniform
distribution as well as return a larger subset of primes (RT81871
<https://rt.cpan.org/Ticket/Display.html?id=81871>). MPU does not depend on Math::Pari though can run
slow for bigints unless the Math::BigInt::GMP or Math::BigInt::Pari modules are installed. Having
Math::Prime::Util::GMP installed also helps performance for MPU. Crypt::Primes is hardcoded to use
Crypt::Random, while MPU uses Bytes::Random::Secure, and also allows plugging in a random function. This
is more flexible, faster, has fewer dependencies, and uses a CSPRNG for security. MPU can return a
primality certificate. What Crypt::Primes has that MPU does not is the ability to return a generator.
Math::Factor::XS calculates prime factors and factors, which correspond to the "factor" and "divisors"
functions of MPU. These functions do not support bigints. Both are implemented with trial division,
meaning they are very fast for really small values, but quickly become unusably slow (factoring 19 digit
semiprimes is over 700 times slower). The function "count_prime_factors" can be done in MPU using
"scalar factor($n)". MPU has no equivalent to "matches", but see the "EXAMPLES" section for a way to
produce the results.
Math::Big version 1.12 includes "primes" functionality. The current code is only usable for very tiny
inputs as it is incredibly slow and uses lots of memory. RT81986
<https://rt.cpan.org/Ticket/Display.html?id=81986> has a patch to make it run much faster and use much
less memory. Since it is in pure Perl it will still run quite slow compared to MPU.
Math::Big::Factors supports factorization using wheel factorization (smart trial division). It supports
bigints. Unfortunately it is extremely slow on any input that isn't the product of just small factors.
Even 7 digit inputs can take hundreds or thousands of times longer to factor than MPU or
Math::Factor::XS. 19-digit semiprimes will take hours versus MPU's single milliseconds.
Math::Factoring is a placeholder module for bigint factoring. Version 0.02 only supports trial division
(the Pollard-Rho method does not work).
Math::Prime::TiedArray allows random access to a tied primes array, almost identically to what MPU
provides in Math::Prime::Util::PrimeArray. MPU has attempted to fix Math::Prime::TiedArray's shift bug
(RT58151 <https://rt.cpan.org/Ticket/Display.html?id=58151>). MPU is typically much faster and will use
less memory, but there are some cases where MP:TA is faster (MP:TA stores all entries up to the largest
request, while MPU:PA stores only a window around the last request).
Math::Primality supports "is_prime", "is_pseudoprime", "is_strong_pseudoprime",
"is_strong_lucas_pseudoprime", "next_prime", "prev_prime", "prime_count", and "is_aks_prime"
functionality. This is a great little module that implements primality functionality. It was the first
CPAN module to support the BPSW test. All inputs are processed using GMP, so it of course supports
bigints. In fact, Math::Primality was made originally with bigints in mind, while MPU was originally
targeted to native integers, but both have added better support for the other. The main differences are
extra functionality (MPU has more functions) and performance. With native integer inputs, MPU is
generally much faster, especially with "prime_count". For bigints, MPU is slower unless the
Math::Prime::Util::GMP module is installed, in which case MPU is ~2x faster. Math::Primality also
installs a "primes.pl" program, but it has much less functionality than the one included with MPU.
Math::NumSeq does not have a one-to-one mapping between functions in MPU, but it does offer a way to get
many similar results such as primes, twin primes, Sophie-Germain primes, lucky primes, moebius, divisor
count, factor count, Euler totient, primorials, etc. Math::NumSeq is set up for accessing these values
in order rather than for arbitrary values, though a few sequences support random access. The primary
advantage I see is the uniform access mechanism for a lot of sequences. For those methods that overlap,
MPU is usually much faster. Importantly, most of the sequences in Math::NumSeq are limited to 32-bit
indices.
Math::Pari supports a lot of features, with a great deal of overlap. In general, MPU will be faster for
native 64-bit integers, while it's differs for bigints (Pari will always be faster if
Math::Prime::Util::GMP is not installed; with it, it varies by function). Note that Pari extends many of
these functions to other spaces (Gaussian integers, complex numbers, vectors, matrices, polynomials,
etc.) which are beyond the realm of this module. Some of the highlights:
"isprime"
The default Math::Pari is built with Pari 2.1.7. This uses 10 M-R tests with randomly chosen bases
(fixed seed, but doesn't reset each invocation like GMP's "is_probab_prime"). This has a greater
chance of false positives compared to the BPSW test. Calling with "isprime($n,1)" will perform a
Pocklington-Lehmer "n-1" proof, but this becomes unreasonably slow past 70 or so digits.
If Math::Pari is built using Pari 2.3.5 (this requires manual configuration) then the primality tests
are completely different. Using "ispseudoprime" will perform a BPSW test and is quite a bit faster
than the older test. "isprime" now does an APR-CL proof (fast, but no certificate).
Math::Primality uses a strong BPSW test, which is the standard BPSW test based on the 1980 paper. It
has no known counterexamples (though like all these tests, we know some exist). Pari 2.3.5 (and
through at least 2.6.2) uses an almost-extra-strong BPSW test for its "ispseudoprime" function. This
is deterministic for native integers, and should be excellent for bigints, with a slightly lower
chance of counterexamples than the traditional strong test. Math::Prime::Util uses the full extra-
strong BPSW test, which has an even lower chance of counterexample. With Math::Prime::Util::GMP,
"is_prime" adds 1 to 5 extra M-R tests using random bases, which further reduces the probability of a
composite being allowed to pass.
"primepi"
Only available with version 2.3 of Pari. Similar to MPU's "prime_count" function in API, but uses a
naive counting algorithm with its precalculated primes, so is not of practical use. Incidently, Pari
2.6 (not usable from Perl) has fixed the pre-calculation requirement so it is more useful, but is
still thousands of times slower than MPU.
"primes"
Doesn't support ranges, requires bumping up the precalculated primes for larger numbers, which means
knowing in advance the upper limit for primes. Support for numbers larger than 400M requires using
Pari version 2.3.5. If that is used, sieving is about 2x faster than MPU, but doesn't support
segmenting.
"factorint"
Similar to MPU's "factor_exp" though with a slightly different return. MPU offers "factor" for a
linear array of prime factors where
n = p1 * p2 * p3 * ... as (p1,p2,p3,...) and "factor_exp" for an array of factor/exponent pairs
where:
n = p1^e1 * p2^e2 * ... as ([p1,e1],[p2,e2],...) Pari/GP returns an array similar to the latter.
Math::Pari returns a transposed matrix like:
n = p1^e1 * p2^e2 * ... as ([p1,p2,...],[e1,e2,...]) Slower than MPU for all 64-bit inputs on an
x86_64 platform, it may be faster for large values on other platforms. With the newer
Math::Prime::Util::GMP releases, bigint factoring is slightly faster on average in MPU.
"divisors"
Similar to MPU's "divisors".
"forprime", "forcomposite", "fordiv", "sumdiv"
Similar to MPU's "forprimes", "forcomposites", "fordivisors", and "divisor_sum".
"eulerphi", "moebius"
Similar to MPU's "euler_phi" and "moebius". MPU is 2-20x faster for native integers. MPU also
supported range inputs, which can be much more efficient. Without Math::Prime::Util::GMP installed,
MPU is very slow with bigints. With it installed, it is about 2x slower than Math::Pari.
"gcd", "lcm", "kronecker", "znorder", "znprimroot"
Similar to MPU's "gcd", "lcm", "kronecker", "znorder", and "znprimroot". Pari's "znprimroot" only
returns the smallest root for prime powers. The behavior is undefined when the group is not cyclic
(sometimes it throws an exception, sometimes it returns an incorrect answer). MPU's "znprimroot"
will always return the smallest root if it exists, and "undef" otherwise.
"sigma"
Similar to MPU's "divisor_sum". MPU is ~10x faster for native integers and about 2x slower for
bigints.
"numbpart"
Similar to MPU's "partitions". This function is not in Pari 2.1, which is the default version used
by Math::Pari. With Pari 2.3 or newer, the functions produce identical results, but Pari is much,
much faster.
"eint1"
Similar to MPU's "ExponentialIntegral".
"zeta"
MPU has "RiemannZeta" which takes non-negative real inputs, while Pari's function supports negative
and complex inputs.
Overall, Math::Pari supports a huge variety of functionality and has a sophisticated and mature code base
behind it (noting that the default version of Pari used is about 10 years old now). For native integers
often using Math::Pari will be slower, but bigints are often superior and it rarely has any performance
surprises. Some of the unique features MPU offers include super fast prime counts, nth_prime, ECPP
primality proofs with certificates, approximations and limits for both, random primes, fast Mertens
calculations, Chebyshev theta and psi functions, and the logarithmic integral and Riemann R functions.
All with fairly minimal installation requirements.
PERFORMANCE
First, for those looking for the state of the art non-Perl solutions:
Primality testing
For general numbers smaller than 2000 or so digits, I believe MPU is the fastest solution (it is
faster than Pari 2.6.2 and PFGW), though FLINT might be a little faster for native sizes. For large
inputs, PFGW <http://sourceforge.net/projects/openpfgw/> is the fastest primality testing software
I'm aware of. It has fast trial division, and is especially fast on many special forms. It does not
have a BPSW test however, and there are quite a few counterexamples for a given base of its PRP test,
so for primality testing it is most useful for fast filtering of very large candidates. A test such
as the BPSW test in this module is then recommended.
Primality proofs
Primo <http://www.ellipsa.eu/> is the best method for open source primality proving for inputs over
1000 digits. Primo also does well below that size, but other good alternatives are WraithX APRCL
<http://sourceforge.net/projects/mpzaprcl/>, the APRCL from the modern Pari <http://pari.math.u-
bordeaux.fr/> package, or the standalone ECPP from this module with large polynomial set.
Factoring
yafu <http://sourceforge.net/projects/yafu/>, msieve <http://sourceforge.net/projects/msieve/>, and
gmp-ecm <http://ecm.gforge.inria.fr/> are all good choices for large inputs. The factoring code in
this module (and all other CPAN modules) is very limited compared to those.
Primes
primesieve <http://code.google.com/p/primesieve/> and yafu <http://sourceforge.net/projects/yafu/>
are the fastest publically available code I am aware of. Primesieve will additionally take advantage
of multiple cores with excellent efficiency. Tomas Oliveira e Silva's private code may be faster for
very large values, but isn't available for testing.
Note that the Sieve of Atkin is not faster than the Sieve of Eratosthenes when both are well
implemented. The only Sieve of Atkin that is even competitive is Bernstein's super optimized
primegen, which runs on par with the SoE in this module. The SoE's in Pari, yafu, and primesieve are
all faster.
Prime Counts and Nth Prime
Outside of private research implementations doing prime counts for "n > 2^64", this module should be
close to state of the art in performance, and supports results up to "2^64". Further performance
improvements are planned, as well as expansion to larger values.
The fastest solution for small inputs is a hybrid table/sieve method. This module does this for
values below 60M. As the inputs get larger, either the tables have to grow exponentially or speed
must be sacrificed. Hence this is not a good general solution for most uses.
PRIME COUNTS
Counting the primes to "800_000_000" (800 million):
Time (s) Module Version Notes
--------- -------------------------- ------- -----------
0.002 Math::Prime::Util 0.35 using extended LMO
0.007 Math::Prime::Util 0.12 using Lehmer's method
0.27 Math::Prime::Util 0.17 segmented mod-30 sieve
0.39 Math::Prime::Util::PP 0.24 Perl (Lehmer's method)
0.9 Math::Prime::Util 0.01 mod-30 sieve
2.9 Math::Prime::FastSieve 0.12 decent odd-number sieve
11.7 Math::Prime::XS 0.26 needs some optimization
15.0 Bit::Vector 7.2
48.9 Math::Prime::Util::PP 0.14 Perl (fastest I know of)
170.0 Faster Perl sieve (net) 2012-01 array of odds
548.1 RosettaCode sieve (net) 2012-06 simplistic Perl
3048.1 Math::Primality 0.08 Perl + Math::GMPz
>20000 Math::Big 1.12 Perl, > 26GB RAM used
Python's standard modules are very slow: MPMATH v0.17 "primepi" takes 169.5s and 25+ GB of RAM. SymPy
0.7.1 "primepi" takes 292.2s. However there are very fast solutions written by Robert William Hanks
(included in the xt/ directory of this distribution): pure Python in 12.1s and NUMPY in 2.8s.
PRIMALITY TESTING
Small inputs: is_prime from 1 to 20M
2.6s Math::Prime::Util (sieve lookup if prime_precalc used)
3.4s Math::Prime::FastSieve (sieve lookup)
4.4s Math::Prime::Util (trial + deterministic M-R)
10.9s Math::Prime::XS (trial)
36.5s Math::Pari w/2.3.5 (BPSW)
78.2s Math::Pari (10 random M-R)
501.3s Math::Primality (deterministic M-R)
Large native inputs: is_prime from 10^16 to 10^16 + 20M
7.0s Math::Prime::Util (BPSW)
42.6s Math::Pari w/2.3.5 (BPSW)
144.3s Math::Pari (10 random M-R)
664.0s Math::Primality (BPSW)
30 HRS Math::Prime::XS (trial)
These inputs are too large for Math::Prime::FastSieve.
bigints: is_prime from 10^100 to 10^100 + 0.2M
2.5s Math::Prime::Util (BPSW + 1 random M-R)
3.0s Math::Pari w/2.3.5 (BPSW)
12.9s Math::Primality (BPSW)
35.3s Math::Pari (10 random M-R)
53.5s Math::Prime::Util w/o GMP (BPSW)
94.4s Math::Prime::Util (n-1 or ECPP proof)
102.7s Math::Pari w/2.3.5 (APR-CL proof)
• MPU is consistently the fastest solution, and performs the most stringent probable prime tests on
bigints.
• Math::Primality has a lot of overhead that makes it quite slow for native size integers. With
bigints we finally see it work well.
• Math::Pari build with 2.3.5 not only has a better primality test, but runs faster. It still has
quite a bit of overhead with native size integers. Pari/gp 2.5.0's takes 11.3s, 16.9s, and 2.9s
respectively for the tests above. MPU is still faster, but clearly the time for native integers is
dominated by the calling overhead.
FACTORING
Factoring performance depends on the input, and the algorithm choices used are still being tuned.
Math::Factor::XS is very fast when given input with only small factors, but it slows down rapidly as the
smallest factor increases in size. For numbers larger than 32 bits, Math::Prime::Util can be 100x or
more faster (a number with only very small factors will be nearly identical, while a semiprime with large
factors will be the extreme end). Math::Pari is much slower with native sized inputs, probably due to
calling overhead. For bigints, the Math::Prime::Util::GMP module is needed or performance will be far
worse than Math::Pari. With the GMP module, performance is pretty similar from 20 through 70 digits,
which the caveat that the current MPU factoring uses more memory for 60+ digit numbers.
This slide presentation <http://math.boisestate.edu/~liljanab/BOISECRYPTFall09/Jacobsen.pdf> has a lot of
data on 64-bit and GMP factoring performance I collected in 2009. Assuming you do not know anything
about the inputs, trial division and optimized Fermat or Lehman work very well for small numbers (<= 10
digits), while native SQUFOF is typically the method of choice for 11-18 digits (I've seen claims that a
lightweight QS can be faster for 15+ digits). Some form of Quadratic Sieve is usually used for inputs in
the 19-100 digit range, and beyond that is the General Number Field Sieve. For serious factoring, I
recommend looking at 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/>, and Pari <http://pari.math.u-bordeaux.fr/>. The latest yafu
should cover most uses, with GGNFS likely only providing a benefit for numbers large enough to warrant
distributed processing.
PRIMALITY PROVING
The "n-1" proving algorithm in Math::Prime::Util::GMP compares well to the version including in Pari.
Both are pretty fast to about 60 digits, and work reasonably well to 80 or so before starting to take
many minutes per number on a fast computer. Version 0.09 and newer of MPU::GMP contain an ECPP
implementation that, while not state of the art compared to closed source solutions, works quite well.
It averages less than a second for proving 200-digit primes including creating a certificate. Times
below 200 digits are faster than Pari 2.3.5's APR-CL proof. For larger inputs the bottleneck is a
limited set of discriminants, and time becomes more variable. There is a larger set of discriminants on
github that help, with 300-digit primes taking ~5 seconds on average and typically under a minute for
500-digits. For primality proving with very large numbers, I recommend Primo <http://www.ellipsa.eu/>.
RANDOM PRIME GENERATION
Seconds per prime for random prime generation on a circa-2009 workstation, with Math::BigInt::GMP,
Math::Prime::Util::GMP, and Math::Random::ISAAC::XS installed.
bits random +testing rand_prov Maurer CPMaurer
----- -------- -------- --------- -------- --------
64 0.0001 +0.000008 0.0002 0.0001 0.022
128 0.0020 +0.00023 0.011 0.063 0.057
256 0.0034 +0.0004 0.058 0.13 0.16
512 0.0097 +0.0012 0.28 0.28 0.41
1024 0.060 +0.0060 0.65 0.65 2.19
2048 0.57 +0.039 4.8 4.8 10.99
4096 6.24 +0.25 31.9 31.9 79.71
8192 58.6 +1.61 234.0 234.0 947.3
random = random_nbit_prime (results pass BPSW)
random+ = additional time for 3 M-R and a Frobenius test
rand_prov = random_proven_prime
maurer = random_maurer_prime
CPMaurer = Crypt::Primes::maurer
"random_nbit_prime" is reasonably fast, and for most purposes should suffice. For cryptographic
purposes, one may want additional tests or a proven prime. Additional tests are quite cheap, as shown by
the time for three extra M-R and a Frobenius test. At these bit sizes, the chances a composite number
passes BPSW, three more M-R tests, and a Frobenius test is extraordinarily small.
"random_proven_prime" provides a randomly selected prime with an optional certificate, without specifying
the particular method. Below 512 bits, using "is_provable_prime"("random_nbit_prime") is typically
faster than Maurer's algorithm, but becomes quite slow as the bit size increases. This leaves the
decision of the exact method of proving the result to the implementation.
"random_maurer_prime" constructs a provable prime. A primality test is run on each intermediate, and it
also constructs a complete primality certificate which is verified at the end (and can be returned).
While the result is uniformly distributed, only about 10% of the primes in the range are selected for
output. This is a result of the FastPrime algorithm and is usually unimportant.
"maurer" in Crypt::Primes times are included for comparison. It is pretty fast for small sizes but gets
slow as the size increases. It does not perform any primality checks on the intermediate results or the
final result (I highly recommended you run a primality test on the output). Additionally important for
servers, "maurer" in Crypt::Primes uses excessive system entropy and can grind to a halt if "/dev/random"
is exhausted (it can take days to return). The times above are on a machine running HAVEGED
<http://www.issihosts.com/haveged/> so never waits for entropy. Without this, the times would be much
higher.
AUTHORS
Dana Jacobsen <dana@acm.org>
ACKNOWLEDGEMENTS
Eratosthenes of Cyrene provided the elegant and simple algorithm for finding primes.
Terje Mathisen, A.R. Quesada, and B. Van Pelt all had useful ideas which I used in my wheel sieve.
Tomas Oliveira e Silva has released the source for a very fast segmented sieve. The current
implementation does not use these ideas. Future versions might.
The SQUFOF implementation being used is a slight modification to the public domain racing version written
by Ben Buhrow. Enhancements with ideas from Ben's later code as well as Jason Papadopoulos's public
domain implementations are planned for a later version.
The LMO implementation is based on the 2003 preprint from Christian Bau, as well as the 2006 paper from
Tomas Oliveira e Silva. I also want to thank Kim Walisch for the many discussions about prime counting.
REFERENCES
• Henri Cohen, "A Course in Computational Algebraic Number Theory", Springer, 1996. Practical
computational number theory from the team lead of Pari <http://pari.math.u-bordeaux.fr/>. Lots of
explicit algorithms.
• Hans Riesel, "Prime Numbers and Computer Methods for Factorization", Birkh?user, 2nd edition, 1994.
Lots of information, some code, easy to follow.
• Pierre Dusart, "Estimates of Some Functions Over Primes without R.H.", preprint, 2010. Updates to
the best non-RH bounds for prime count and nth prime. <http://arxiv.org/abs/1002.0442/>
• Pierre Dusart, "Autour de la fonction qui compte le nombre de nombres premiers", PhD thesis, 1998.
In French. The mathematics is readable and highly recommended reading if you're interesting in prime
number bounds. <http://www.unilim.fr/laco/theses/1998/T1998_01.html>
• Gabriel Mincu, "An Asymptotic Expansion", Journal of Inequalities in Pure and Applied Mathematics,
v4, n2, 2003. A very readable account of Cipolla's 1902 nth prime approximation.
<http://www.emis.de/journals/JIPAM/images/153_02_JIPAM/153_02.pdf>
• Christian Bau, "The Extended Meissel-Lehmer Algorithm", 2003, preprint with example C++
implementation. Very detailed implementation-specific paper which was used for the implementation
here. Highly recommended for implementing a sieve-based LMO.
<http://cs.swan.ac.uk/~csoliver/ok-sat-library/OKplatform/ExternalSources/sources/NumberTheory/ChristianBau/>
• David M. Smith, "Multiple-Precision Exponential Integral and Related Functions", ACM Transactions on
Mathematical Software, v37, n4, 2011. <http://myweb.lmu.edu/dmsmith/toms2011.pdf>
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Integral", Journal of Graphics, GPU, and Game Tools, v15, n3, pp 183-198, 2011.
<http://www.cs.utah.edu/~vpegorar/research/2011_JGT/paper.pdf>
• William H. Press et al., "Numerical Recipes", 3rd edition.
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Mathematics of Computation, v23, pp 289-303, 1969.
<http://www.ams.org/journals/mcom/1969-23-106/S0025-5718-1969-0242349-2/>
• W. J. Cody and Henry C. Thacher, Jr., "Rational Chebyshev Approximations for the Exponential Integral
E_1(x)", Mathematics of Computation, v22, pp 641-649, 1968.
• W. J. Cody, K. E. Hillstrom, and Henry C. Thacher Jr., "Chebyshev Approximations for the Riemann Zeta
Function", "Mathematics of Computation", v25, n115, pp 537-547, July 1971.
• Ueli M. Maurer, "Fast Generation of Prime Numbers and Secure Public-Key Cryptographic Parameters",
1995. Generating random provable primes by building up the prime.
<http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.26.2151>
• Pierre-Alain Fouque and Mehdi Tibouchi, "Close to Uniform Prime Number Generation With Fewer Random
Bits", pre-print, 2011. Describes random prime distributions, their algorithm for creating random
primes using few random bits, and comparisons to other methods. Definitely worth reading for the
discussions of uniformity. <http://eprint.iacr.org/2011/481>
• Douglas A. Stoll and Patrick Demichel , "The impact of X(s) complex zeros on X(x) for x <
10^{10^{13}}", "Mathematics of Computation", v80, n276, pp 2381-2394, October 2011.
<http://www.ams.org/journals/mcom/2011-80-276/S0025-5718-2011-02477-4/home.html>
• OEIS: Primorial <http://oeis.org/wiki/Primorial>
• Walter M. Lioen and Jan van de Lune, "Systematic Computations on Mertens' Conjecture and Dirichlet's
Divisor Problem by Vectorized Sieving", in From Universal Morphisms to Megabytes, Centrum voor
Wiskunde en Informatica, pp. 421-432, 1994. Describes a nice way to compute a range of Moebius
values. <http://walter.lioen.com/papers/LL94.pdf>
• Marc Deleglise and Jooel Rivat, "Computing the summation of the Moebius function", Experimental
Mathematics, v5, n4, pp 291-295, 1996. Enhances the Moebius computation in Lioen/van de Lune, and
gives a very efficient way to compute the Mertens function.
<http://projecteuclid.org/euclid.em/1047565447>
• Manuel Benito and Juan L. Varona, "Recursive formulas related to the summation of the Moebius
function", The Open Mathematics Journal, v1, pp 25-34, 2007. Among many other things, shows a simple
formula for computing the Mertens functions with only n/3 Moebius values (not as fast as Deleglise
and Rivat, but really simple).
<http://www.unirioja.es/cu/jvarona/downloads/Benito-Varona-TOMATJ-Mertens.pdf>
• 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>
COPYRIGHT
Copyright 2011-2014 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.