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Module Version: 0.18   Source   Latest Release: Math-Prime-Util-0.26

# NAME

Math::Prime::Util - Utilities related to prime numbers, including fast sieves and factoring

Version 0.18

# 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 )};

# For non-bigints, is_prime and is_prob_prime will always be 0 or 2.
# They return 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);

# Strong Lucas-Selfridge test
say "\$n is a prime or slpsp" if is_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 );

# Get all factors
@divisors = all_factors( \$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";

# divisor sum
\$sigma = divisor_sum( \$n, sub { \$_[0] } );

# The primorial n# (product of all primes <= n)
say "15# (2*3*5*7*11*13) is ", primorial(15);
# The primorial p(n)# (product of first n primes)
say "P(9)# (2*3*5*7*11*13*17*19*23) is ", pn_primorial(9);

# 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

# 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. It requires no external software for big number support, as there are Perl implementations included that solely use Math::BigInt and Math::BigFloat. However, performance will be improved for most big number functions by installing Math::Prime::Util::GMP, and is definitely recommended if you do many bignum operations. Also look into Math::Pari as an alternative.

The module is thread-safe and allows concurrency between Perl threads while still sharing a prime cache. It is not itself multithreaded. See the Limitations section if you are using Win32 and threads in your program.

# BIGNUM SUPPORT

By default all functions support bignums. With a few exceptions, the module will not turn on bignum support for you -- you will need to `use bigint`, `use bignum`, or pass in a Math::BigInt or Math::BigFloat object as your input. The functions take some care to perform all bignum operations using the same class as was passed in, allowing the module to work properly with Calc, FastCalc, GMP, Pari, etc. You should try to install Math::Prime::Util::GMP if you plan to use bigints with this module, as it will make it run much faster.

Some of the functions, notably:

```  factor
is_prime
is_prob_prime
is_strong_pseudoprime
next_prime
prev_prime
nth_prime```

work very fast (under 1 microsecond) on small inputs, but the wrappers for input validation and bigint support take more time than the function itself. Using the flag '-bigint', e.g.:

`  use Math::Prime::Util qw(-bigint);`

will turn off bigint support for those functions. Those functions will then go directly to the XS versions, which will speed up very small inputs a lot. This is useful if you're using the functions in a loop, but since the difference is less than a millisecond, it's really not important in general (also, a future implementation may find a way to speed this up without the option).

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 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.
Having run these functions on many versions of Perl, 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.

# FUNCTIONS

## is_prime

`  print "\$n is prime" if is_prime(\$n);`

Returns 2 if the number is prime, 0 if not. For numbers larger than `2^64` it will return 0 for composite and 1 for probably prime, using a strong BPSW test. If Math::Prime::Util::GMP is installed, some quick primality proofs are run on larger numbers, so will return 2 for many of those also.

Also see the "is_prob_prime" function, which will never do additional tests, and the "is_provable_prime" function which will try very hard to return only 0 or 2 for any input.

For native precision numbers (anything smaller than `2^64`, all three functions are identical and use a deterministic set of Miller-Rabin tests. While "is_prob_prime" and "is_prime" return probable prime results for larger numbers, they use the strong Baillie-PSW test, which has had no counterexample found since it was published in 1980 (though certainly they exist).

## 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 trial division (for ranges with only one expected prime), 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. If the input is not a bigint, then 0 is returned if the next prime is larger than a native integer type (the last representable primes being `4,294,967,291` in 32-bit Perl and `18,446,744,073,709,551,557` in 64-bit).

## prev_prime

`  \$n = prev_prime(\$n);`

Returns the prime smaller than the input number. 0 is returned if the input is `2` or lower.

## 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, and `14,16` returns 0).

The current implementation decides based on the ranges whether to use a segmented sieve with a fast bit count, or Lehmer's algorithm. The former is prefered 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`.

Lehmer's method has complexity approximately `O(b^0.7) + O(a^0.7)`. It does use more memory however. A calculation of `Pi(10^14)` completes in under 1 minute, `Pi(10^15)` in under 5 minutes, and `Pi(10^16)` in under 30 minutes, however using nearly 1400MB of peak memory for the last. In contrast, even primesieve using 12 cores would take over a week on this same computer 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_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. The current implementation uses the Riemann R function which is quite accurate: an error of less than `0.0005%` is typical for input values over `2^32`. A slightly faster (0.1ms vs. 1ms) 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 2 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, the Dusart 2010 bounds are calculated, Lehmer's fast prime counting method is used to calculate the count up to that point, then sieving is done in the range between the bounds.

While this method is hundreds of times faster than generating primes, and doesn't involve big tables of precomputed values, it still can take a fair amount of time and space for large inputs. Calculating the `10^11th` prime takes a bit over 2 seconds, the `10^12th` prime takes 20 seconds, and the `10^13th` prime (323780508946331) takes 4 minutes. Think about whether a bound or approximation would be acceptable, as they can be computed analytically.

If the bigint or bignum module is not in use, this will generate an overflow exception if the number requested would result in a prime that cannot fit in a native type. If bigints are in use, then the calculation will proceed, though it will be exceedingly slow. A later version of Math::Prime::Util::GMP may include this functionality which would help for 32-bit machines.

## 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_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 prime or a strong pseudoprime 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 numbers 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 being deprecated.

## is_strong_lucas_pseudoprime

Takes a positive number as input, and returns 1 if the input is a strong Lucas pseudoprime 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).

## 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 a tuned set of Miller-Rabin tests such that the result will be deterministic. Either 2, 3, 4, 5, or 7 Miller-Rabin tests are performed (no more than 3 for 32-bit input), and the result will then always be 0 (composite) or 2 (prime). A later implementation may change the internals, but the results will be identical.

For inputs larger than `2^64`, a strong Baillie-PSW primality test is performed (aka 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_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`.

The current implementation uses a Lucas test requiring a complete factorization of `n-1`, which may not be possible in a reasonable amount of time. The GMP version uses the BLS (Brillhart-Lehmer-Selfridge) method, requiring `n-1` to be factored to the cube root of `n`, which is more likely to succeed and will usually take less time, but can still fail. Hence you should always test that the result is `2` to ensure the prime is proven.

## 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.

This function is only included for completeness and as an example. While the implementation is fast compared to the only other Perl implementation available (in Math::Primality), it is slow compared to others. However, even optimized AKS implementations are far slower than ECPP or other modern primality tests.

## moebius

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

Returns the Möbius function (also called the Moebius, Mobius, or MoebiusMu function) for a positive non-zero 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.

## euler_phi

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

Returns the Euler totient function (also called Euler's phi or phi function) for an integer value. This is an arithmetic function that 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. Mathematic/WolframAlpha also returns 0 for input 0, but returns `euler_phi(-n)` for `n < 0`.

## 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)`.

## divisor_sum

`  say "Sum of divisors of \$n:", divisor_sum( \$n, sub { \$_[0] } );`

This function takes a positive integer as input, along with a code reference. For each positive divisor of the input, including 1 and itself, the coderef is called with the divisor as the only argument, and the return values summed. There are a number of utilities this can be used for, though it may not be the most efficient way to calculate them. Example:

`  divisor_sum( \$n, sub { my \$d=shift; \$d**5 * moebius(\$n/\$d); } );`

calculates the 5th Jordan totient (OEIS 59378). In this example we have a specific function `jordan_totient` that can compute this more efficiently.

`  divisor_sum( \$n, sub { \$_[0] ** \$k } );`

calculates sigma_k (OEIS A000005, A000203, A001157, A001158 for k=0..3). The simple sigma shown as the first example can be used to find aliquot sums, abundant numbers, perfect numbers, and more.

## 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: 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.

## 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: 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.

## 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 psuedo-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 rand function is called one or more times for selection.

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.

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 `rand` may 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. 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 some other modules behave.

If no `irand` function was set, then we check if the sub `main::rand` exists and use it if so. It will be called with no arguments and should return a floating point value in the interval [0,1) with 32 bits of entropy. This allows the `rand` function from most CPAN modules to be used.

Lastly, if no `irand` function was set and no sub <main::rand> exists, then the system rand will be used. System rand functions are notoriously poor, and a later version of this module may implement something like TinyMT to cover the default case.

Examples of irand functions:

```  # Math::Random::Secure.  Uses ISAAC and strong seed methods.  Recommended.
use Math::Random::Secure;
prime_set_config(irand => \&Math::Random::Secure::irand);

# Bytes::Random::Secure.  Also uses ISAAC and strong seed methods.
use Bytes::Random::Secure qw/random_bytes/;
prime_set_config(irand => sub { return unpack("L", random_bytes(4)); });

# 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});```

Examples of main::rand, where this is done in your script:

```  use Math::Random::Secure qw/rand/;

use Bytes::Random::Secure qw/random_bytes/;
sub rand { (\$_[0]||1) * (unpack("L", random_bytes(4))/4294967296.0)}

use Math::Random::MT::Auto qw/rand/;

sub rand { ... do your own cool stuff here ... }```

For cryptographically secure primes, you need to use something better than the default for both seeding and random number generation. I would recommend using Math::Random::Secure and also installing Math::Random::ISAAC::XS if possible. It is reasonably fast and does everything needed by default. For more information, I recommend reading the documentation for Math::Random::Secure and Bytes::Random::Secure.

## 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 between 1 and the maximum native type (10 for 32-bit, 20 for 64-bit, 10000 if bigint is active). One of the primes within that range (e.g. 1000 - 9999 for 4-digits) will be uniformly selected using the rand 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' tag was used, then numbers larger than the threshold will be flagged as an error, and numbers on the threshold will be restricted to native numbers.

## random_nbit_prime

`  my \$bigprime = random_nbit_prime(512);`

Selects a random n-bit prime, where the input is an integer number of bits between 2 and the maximum representable bits (32, 64, or 100000 for native 32-bit, native 64-bit, and bigint respectively). A prime with the nth bit set will be uniformly selected, with randomness supplied via calls to the rand function as described above.

Since this uses the random_prime function, all uniformity properties of that function apply to this. The n-bit range is partitioned into nearly equal segments less than `2^32`, a segment is randomly selected, then the trivial Monte Carlo algorithm is used to select a prime from within the segment. This gives a reasonably uniform distribution, doesn't use excessive random source, and can be very fast.

The result will be a BigInt if the number of bits is greater than the native bit size. For better performance with very large bit sizes, install Math::BigInt::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 uses 128 as a minimum.
• 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 using strong primes.

## 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 cryptographic purposes you need to ensure you're using a good RNG that is well seeded. See the notes for "random_prime".

The differences between this function and that in Crypt::Primes include

• Version 0.50 of Crypt::Primes can return composites.
• Version 0.50 of Crypt::Primes uses the `PRIMEINC` algorithm for the base generator, which gives a very non-uniform distribution. This differs from Maurer's algorithm which uses the Monte Carlo algorithm (which is what this module uses).
• No external libraries are needed for this module, while C::P requires Math::Pari. See the next item however.
• Crypt::Primes is quite fast for all sizes since it uses Pari for all heavy lifting. M::P::U is really fast for native bit sizes. It is similar speed to Crypt::Primes if the BigInt package in use is GMP or Pari, e.g.
`   use Math::BigInt lib=>'GMP';`

but a lot slower without. Having the Math::Prime::Util::GMP module installed helps in any case.

• Crypt::Primes has some useful options for cryptography.
• Crypt::Primes is hardcoded to use Crypt::Random, while M::P::U allows plugging in the random function. This is more flexible but also prone to misuse. You ought to use something like Math::Random::Secure.

Any feedback on this function would be greatly appreciated.

# 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, and is the preferred way to use
cryptographic RNGs.```

# 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 special cases of `n = 0` and `n = 1` will return `n`, which guarantees multiplying the factors together will always result in the input value, though those are the only cases where the returned factors are not prime.

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.

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).

## all_factors

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

Produces all the divisors of a positive number input. 1 and the input number are excluded (which implies that an empty list is returned for any prime number input). The divisors are a power set of multiplications of the prime factors, returned as a uniqued sorted list.

## trial_factor

`  my @factors = trial_factor(\$n);`

Produces the prime factors of a positive number input. The factors will be in numerical order. The special cases of `n = 0` and `n = 1` will return `n`, while with all other inputs the factors are guaranteed to be prime. 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.

## rsqufof_factor

```  my @factors = squfof_factor(\$n);
my @factors = rsqufof_factor(\$n);  # racing multiplier version```

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.

## 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).

# 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 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).

MPFR is used for positive inputs only. If Math::MPFR is not installed 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 terminilogy 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 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).

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 ζ(s)-1, where ζ(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 larger 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 there are defects in Math::BigFloat with high accuracy computations that make this difficult. 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 pseudoprimes base 17:

`    perl -MMath::Prime::Util=:all -E 'my \$n=\$base|1; while(1) { print "\$n " if is_strong_pseudoprime(\$n,\$base) && !is_prime(\$n); \$n+=2; } BEGIN {\$|=1; \$base=17}'`

Print some primes above 64-bit range:

```    perl -MMath::Prime::Util=:all -Mbigint -E 'my \$start=100000000000000000000; say join "\n", @{primes(\$start,\$start+1000)}'
# Similar code using 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); }'```

# LIMITATIONS

I have not completed testing all the functions near the word size limit (e.g. `2^32` for 32-bit machines). Please report any problems you find.

Perl versions earlier than 5.8.0 have issues with 64-bit that show up in the factoring tests. The test suite will try to determine if your Perl is broken. If you use later versions of Perl, or Perl 5.6.2 32-bit, or Perl 5.6.2 64-bit and keep numbers below `~ 2^52`, then everything works. The best solution is to update to a more recent Perl.

The module is thread-safe and should allow good concurrency on all platforms that support Perl threads except Win32 (Cygwin works). 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.

# PERFORMANCE

Counting the primes to `10^10` (10 billion), with time in seconds. Pi(10^10) = 455,052,511. The numbers below are for sieving. Calculating `Pi(10^10)` takes 0.064 seconds using the Lehmer algorithm in version 0.12.

```   External C programs in C / C++:

1.9  primesieve 3.6 forced to use only a single thread
2.2  yafu 1.31
3.8  primegen (optimized Sieve of Atkin, conf-word 8192)
5.6  Tomás Oliveira e Silva's unoptimized segmented sieve v2 (Sep 2010)
6.7  Achim Flammenkamp's prime_sieve (32k segments)
9.3  http://tverniquet.com/prime/ (mod 2310, single thread)
11.2  Tomás Oliveira e Silva's unoptimized segmented sieve v1 (May 2003)
17.0  Pari 2.3.5 (primepi)

Small portable functions suitable for plugging into XS:

4.1  My segmented SoE used in this module (with unrolled inner loop)
15.6  My Sieve of Eratosthenes using a mod-30 wheel
17.2  A slightly modified verion of Terje Mathisen's mod-30 sieve
35.5  Basic Sieve of Eratosthenes on odd numbers
33.4  Sieve of Atkin, from Praxis (not correct)
72.8  Sieve of Atkin, 10-minute fixup of basic algorithm
91.6  Sieve of Atkin, Wikipedia-like```

Perl modules, counting the primes to `800_000_000` (800 million):

```  Time (s)   Module                      Version  Notes
---------  --------------------------  -------  -----------
0.03  Math::Prime::Util           0.12     using Lehmer's method
0.28  Math::Prime::Util           0.17     segmented mod-30 sieve
0.52  Math::Prime::Util::PP       0.14     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             0.29     "" but needs a count API
15.0   Bit::Vector                 7.2
57.3   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
~11000     Math::Primality             0.04     Perl + Math::GMPz
>20000     Math::Big                   1.12     Perl, > 26GB RAM used```

Python can do this in 2.8s using an RWH numpy function, 14.3s using an RWH pure Python function. However the standard modules are far slower. mpmath v0.17 primepi takes 169.5s and 25+ GB of RAM. sympi 0.7.1 primepi takes 292.2s.

`is_prime`: my impressions for various sized inputs:

```   Module                   1-10 digits  10-20 digits  BigInts
-----------------------  -----------  ------------  --------------
Math::Prime::Util        Very fast    Pretty fast   Slow to Fast (3)
Math::Prime::XS          Very fast    Very slow (1) --
Math::Prime::FastSieve   Very fast    N/A (2)       --
Math::Primality          Very slow    Very slow     Fast
Math::Pari               Slow         OK            Fast

(1) trial division only.  Very fast if every factor is tiny.
(2) Too much memory to hold the sieve (11dig = 6GB, 12dig = ~50GB)
(3) If L<Math::Prime::Util::GMP> is installed, then all three of the
BigInt capable modules run at reasonble similar speeds, capable of
performing the BPSW test on a 3000 digit input in ~ 1 second.  Without
that module all computations are done in Perl, so this module using
GMP bigints runs 2-3x slower, using Pari bigints about 10x slower,
and using the default bigints (Calc) it can run much slower.```

The differences are in the implementations:

Math::Prime::Util looks in the sieve for a fast bit lookup if that exists (default up to 30,000 but it can be expanded, e.g. `prime_precalc`), uses trial division for numbers higher than this but not too large (0.1M on 64-bit machines, 100M on 32-bit machines), a deterministic set of Miller-Rabin tests for 64-bit and smaller numbers, and a BPSW test for bigints.
Math::Prime::XS does trial divisions, which is wonderful if the input has a small factor (or is small itself). But if given a large prime it can take orders of magnitude longer. It does not support bigints.
Math::Prime::FastSieve only works in a sieved range, which is really fast if you can do it (M::P::U will do the same if you call `prime_precalc`). Larger inputs just need too much time and memory for the sieve.
Math::Primality uses GMP for all work. Under ~32-bits it uses 2 or 3 MR tests, while above 4759123141 it performs a BPSW test. This is is fantastic for bigints over 2^64, but it is significantly slower than native precision tests. With 64-bit numbers it is generally an order of magnitude or more slower than any of the others. Once bigints are being used, its performance is quite good. It is faster than this module unless Math::Prime::Util::GMP has been installed, in which case this module is just a little bit faster.
Math::Pari has some very effective code, but it has some overhead to get to it from Perl. That means for small numbers it is relatively slow: an order of magnitude slower than M::P::XS and M::P::Util (though arguably this is only important for benchmarking since "slow" is ~2 microseconds). Large numbers transition over to smarter tests so don't slow down much. The `ispseudoprime(n,0)` function will perform the BPSW test and is fast even for large inputs.

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's underlying algorithms and code are much more mature than this module, and for 21+ digit numbers will be a better choice. Small numbers factor much faster with Math::Prime::Util. For 30+ digit numbers, Math::Pari is much faster. Without the Math::Prime::Util::GMP module, almost all actions on numbers greater than native scalars will be much faster in Pari.

The presentation here: 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, msieve, gmp-ecm, GGNFS, and Pari.

The primality proving algorithms leave much to be desired. If you have numbers larger than `2^128`, I recommend Pari's `isprime(n, 2)` which will run a fast APRCL test, or GMP-ECPP. Either one will be much faster than the Lucas or BLS algorithms used in MPU for large inputs.

# AUTHORS

Dana Jacobsen <dana@acm.org>

# ACKNOWLEDGEMENTS

Eratosthenes of Cyrene provided the elegant and simple algorithm for finding the primes.

Terje Mathisen, A.R. Quesada, and B. Van Pelt all had useful ideas which I used in my wheel sieve.

Tomás 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 my modifications to Ben Buhrow's modifications to Bob Silverman's code. I may experiment with some other implementations (Ben Buhrows and Jason Papadopoulos both have published excellent versions in the public domain).

# REFERENCES

Pierre Dusart, "Estimates of Some Functions Over Primes without R.H.", preprint, 2010. http://arxiv.org/abs/1002.0442/
Pierre Dusart, "Autour de la fonction qui compte le nombre de nombres premiers", PhD thesis, 1998. In French, but 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
David M. Smith, "Multiple-Precision Exponential Integral and Related Functions".
Vincent Pegoraro and Philipp Slusallek, "On the Evaluation of the Complex-Valued Exponential Integral".
William H. Press et al., "Numerical Recipes", 3rd edition.
W. J. Cody and Henry C. Thacher, Jr., "Rational Chevyshev Approximations for the Exponential Integral E_1(x)".
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. 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", 2011. http://eprint.iacr.org/2011/481
Douglas A. Stoll and Patrick Demichel , "The impact of ζ(s) complex zeros on π(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.