GMPz(3p)

NAME
   Math::GMPz - perl interface to the GMP library's integer (mpz) functions.

DEPENDENCIES
   This module needs the GMP C library - available from:
   L<https://gmplib.org>

DESCRIPTION
   A bignum module utilising the Gnu MP (GMP) library.
   Basically this module simply wraps nearly all of
   the integer functions provided by that library.
   The documentation below extensively plagiarises
   the documentation at L<https://gmplib.org>.
   See the Math::GMPz test suite for examples of
   usage.

SYNOPSIS
   use Math::GMPz qw(:mpz);

   my $string = 'fa9eeeeeeeeeeeeea1234dcbaef1';
   my $base = 16;

   # Create the Math::GMPz object
   my $bn1 = Rmpz_init_set_str($string, $base);

   # Create another Math::GMPz object that holds
   # an initial value of zero, but has enough
   # memory allocated to store a 131-bit number.
   # If 131 bits turns out to be insufficient, it
   # doesn't matter - additional memory is allocated
   # automatically to Math::GMPz objects as needed
   # by the GMP library.
   my $bn2 = Rmpz_init2(131);

   # Create another Math::GMPz object initialised to 0.
   my $bn3 = Rmpz_init();

   # or use the new() function:
   my $bn4 = Math::GMPz->new(12345);

   # Perform some operations ... see 'FUNCTIONS' below.

   .
   .

   # print out the value held by $bn1 (in octal):
   print Rmpz_get_str($bn1, 8), "\n";

   # print out the value held by $bn1 (in decimal):
   print Rmpz_get_str($bn1, 10);

   # print out the value held by $bn1 (in base 29)
   # using the (alternative) Rmpz_out_str()
   # function. (This function doesn't print a newline.)
   Rmpz_out_str($bn1, 29);

MEMORY MANAGEMENT
   Objects created with the Rmpz_init* functions have
   been blessed into package Math::GMPz. They will
   therefore be automatically cleaned up by the DESTROY()
   function whenever they go out of scope.

   For each Rmpz_init* function there is a corresponding function
   called Rmpz_init*_nobless which returns an
   unblessed object. If you create Math::GMPz objects
   using the '_nobless' versions, it will then be up to you
   to clean up the memory associated with these objects by
   calling Rmpz_clear($op) for each object. Alternatively
   such objects will be cleaned up when the script ends.
   I don't know why you would want to create unblessed
   objects. The point is that you can if you want to.

FUNCTIONS
   See the GMP documentation at http://gmplib.org.

   These next 3 functions are demonstrated above:
   $rop   = Rmpz_init_set_str($str, $base); # 1 < $base < 37
   $rop   = Rmpz_init2($bits); # $bits > 0
   $str = Rmpz_get_str($r, $base); # 1 < $base < 37

   The following functions are simply wrappers around a GMP
   function of the same name. eg. Rmpz_swap() is a wrapper around
   mpz_swap().

   "$rop", "$op1", "$op2", etc. are Math::GMPz objects - the
   return values of one of the Rmpz_init* functions.
   They are in fact references to GMP structures.
   The "$rop" argument(s) contain the result(s) of the calculation
   being done, the "$op" argument(s) being the input(s) into that
   calculation.
   Generally, $rop, $op1, $op2, etc. can be the same perl variable,
   though usually they will be distinct perl variables
   referencing distinct GMP structures.
   Eg something like Rmpz_add($r1, $r1, $r1),
   where $r1 *is* the same reference to the same GMP structure,
   would add $r1 to itself and store the result in $r1. Think of it
   as $r1 += $r1. Otoh, Rmpz_add($r1, $r2, $r3), where each of the
   arguments is a different reference to a different GMP structure
   would add $r2 to $r3 and store the result in $r1. Think of it as
   $r1 = $r2 + $r3.
   Mostly, the first argument is the argument that
   stores the result and subsequent arguments provide the input values.
   Exceptions to this can be found in some of the functions that
   actually return a value, and, eg., the div_qr functions (which
   yield both quotient and remainder as their first *two* arguments).
   Like I say, see the GMP manual for details. I hope it's
   intuitively obvious or quickly becomes so. Also see the test
   suite that comes with the distro for some examples of usage.

   "$ui" means any integer that will fit into a C 'unsigned long int'.

   "$si" means any integer that will fit into a C 'signed long int'.

   "$double" means any number (not necessarily integer) that will fit
   into a C 'double'.

   "$bool" means a value (usually a 'signed long int') in which
   the only interest is whether it evaluates as true or not.

   "$str" simply means a string of symbols that represent a number,
   eg "1234567890987654321234567" which might be a base 10 number,
   or "zsa34760sdfgq123r5" which would have to represent at least
   a base 36 number (because "z" is a valid digit only in bases 36
   and higher). Valid bases for GMP numbers are 2..62 .

   "$NULL" is $Math::GMPz::NULL (the NULL mpz type).

   #####################

   INITIALIZING INTEGERS

   Normally, a variable should be initialized once only or at least be
   cleared, using `Rmpz_clear', between initializations.
   'DESTROY' (which calls 'Rmpz_clear') is automatically called on
   blessed objects whenever they go out of scope.

   First read the section 'MEMORY MANAGEMENT' (above).

   $rop = Math::GMPz::new();
   $rop = Math::GMPz->new();
   $rop = new Math::GMPz();
   $rop = Rmpz_init();
   $rop = Rmpz_init_nobless();
    Initialize $rop, and set its value to 0.

   $rop = Rmpz_init2($bits);
   $rop = Rmpz_init2_nobless($bits);
    Initialize $rop, with space for $bits bits, and set its value
    to 0. $bits is only the initial space, $rop will grow
    automatically if necessary, for subsequent values stored.
    `Rmpz_init2' makes it possible to avoid such reallocations if a
    maximum size is known in advance.

   Rmpz_realloc2($rop, $ui);
    Change the space allocated for $rop to $ui bits.  The value in
    $rop is preserved if it fits, or is set to 0 if not.

   ##################

   ASSIGNING INTEGERS

   Rmpz_set($rop, $op);
    Assign the value in $op to $rop.

   Rmpz_set_si($rop, $si);
    Assign the 'signed int', $si, to $rop.

   Rmpz_set_ui($rop, $ui);
    Assign the 'unsigned int', $ui, to $rop.

   Rmpz_set_sj($rop, $sj); # GMP does not provide mpz_set_sj
    Assign the 'signed long long int', $sj, to $rop.
    Available only if $Config{ivtype} is 'long long'.

   Rmpz_set_uj($rop, $uj); # GMP does not provide mpz_set_uj
    Assign the 'unsigned long long int', $uj, to $rop.
    Available only if $Config{ivtype} is 'long long'.

   Rmpz_set_d($rop, $double);
    Assign $double to $rop. (First truncate to an integer if $double
    does not contain an integer value.)
    Program dies with an appropriate error message if $double is
    Inf or NaN.

   Rmpz_set_q($rop, $q); # $q is a Math::GMPq or GMP::Mpq object
    Assign $q to $rop. (Truncate to an integer if necessary.)

   Rmpz_set_f($rop, $f); # $f is a Math::GMPf or GMP::Mpf object
    Assign $f to $rop. (Truncate to an integer if necessary.)

   Rmpz_set_str       ($rop, $str, $base);
    Set $rop to the base $base value of $str. $base may vary from
    2 to 62.  If $base is 0, the actual base is determined from the
    leading characters: if the first two characters are "0x" or "0X",
    hexadecimal is assumed, otherwise if the first character is "0",
    octal is assumed, otherwise decimal is assumed.
    If the string is deemed to be a non-valid base $base number, then
    the function croaks with an appropriate error message.

   Rmpz_set_IV($rop, $iv); # $iv is a perl integer value (IV)
    Set $rop to the value of $iv. (If $iv is an unsigned integer
    value then that unsigned value is assigned.)
    $iv must have its IOK flag set, or the function will croak.
    Best to first check IOK_flag($iv), which will return a non-zero
    value if and only if the IOK flag is set.

   Rmpz_set_NV($rop, $nv); # $nv is a perl floating point value (NV)
    Set $rop to the value of $nv. (First truncate to an integer if
    $nv does not contain an integer value.)
    The program dies with an appropriate error message if $nv is
    Inf or NaN.
    $nv must have its NOK flag set, or the function will croak.
    Best to first check NOK_flag($nv), which will return a non-zero
    value if and only if the NOK flag is set.

   Rmpz_swap($rop1, $rop2); # swap the values

   ######################################

   COMBINED INITIALIZATION AND ASSIGNMENT

   Both initialise and set the value in one step.

   NOTE: Do NOT use these functions if $rop has already
   been initialised. Instead use the Rmpz_set* functions
   in 'Assigning Integers' (above)

   First read the section 'MEMORY MANAGEMENT' (above).

   $rop = Math::GMPz->new($arg);
   $rop = Math::GMPz::new($arg);
   $rop = new Math::GMPz($arg);
    Returns a Math::GMPz object with the value of $arg.
    $arg can be either a Math::GMPz object, a Math::GMP object, a
    Math::GMPq object, a Math::MPFR object, a Math::BigInt object,
    a number (IV, UV or NV) or a string that represents an integer value.
    Strings that are written in scientific notation (eg '1e200' or
    '0x9p+40') are not currently accepted. Such arguments could, eg,
    be made "acceptable" if rewritten as '1'.('0'x200) or,
    respectively, '0b1001'.('0'x40).
    If $arg is a string, an optional additional argument that specifies
    the base of the number can be supplied to new(). If base is 0 (or
    not supplied) then the leading characters of the string are used:
    0x or 0X for hex, 0b or 0B for binary, 0 for octal, or decimal
    otherwise. Legal values for the base are 0 and 2..62 .
    For any non-string argument that represents a non-integral value
    (ie NV, Math::GMPq object, or Math::MPFR object), the value is
    first rounded towards zero to an integral value, before being
    assigned to the Math::GMPz object.

   $rop = Rmpz_init_set($op);
   $rop = Rmpz_init_set_nobless($op);
   $rop = Rmpz_init_set_ui($ui);
   $rop = Rmpz_init_set_ui_nobless($ui);
   $rop = Rmpz_init_set_si($si);
   $rop = Rmpz_init_set_si_nobless($si);
   $rop = Rmpz_init_set_d($double);
   $rop = Rmpz_init_set_d_nobless($double);
   $rop = Rmpz_init_set_str($str, $base);
   $rop = Rmpz_init_set_str_nobless($str, $base);

   $rop = Rmpz_init_set_IV($IV); # $IV is a perl integer value
   $rop = Rmpz_init_set_IV_nobless($IV);
   $rop = Rmpz_init_set_NV($NV); # $NV is a perl floating point value
   $rop = Rmpz_init_set_NV_nobless($NV);
    Note that Rmpz_init_set_d and Rmpz_init_set_NV will truncate the
    double/NV to an integer value prior to assigning if the double/NV
    does not hold an integer value.
    Assigning an Inf or NaN is a fatal error.

   $rop = new_from_MBI($mbi); # $mbi is a Math::BigInt object.

   ###################

   CONVERTING INTEGERS

   $ui = Rmpz_get_ui($op);
    Return the value of $op as an `unsigned long'.
    The sign of $op is ignored, only the absolute value is used.
    If $op is either too big to fit into an `unsigned long int', or
    is negative, the returned result is probably not very useful.
    To find out if the value will fit, use the function
    `Rmpz_fits_ulong_p'.

   $si = Rmpz_get_si($op);
    If $op fits into a `signed long int' return the value of $op.
    Otherwise return the least significant part of $op, with the
    same sign as $op. If $op is too big to fit in a `signed long
    int', the returned result is probably not very useful.  To
    find out if the value will fit, use the function
    `Rmpz_fits_slong_p'.

   $IV = Rmpz_get_IV($op);
    If $op fits into either an IV or a UV return the value of $op.
    Otherwise return the least significant part of $op, with the
    same sign as $op. If $op does not fit into either an IV or a
    UV, the returned result is probably not very useful. To find
    find out if the value will fit, use the 'Rmpz_fits_IV_p'
    function.

   $double = Rmpz_get_d($op);
     Assign the value of $op, expressed in double precision, to
     an NV.

   $NV = Rmpz_get_NV($op);
     Assign the value of $op to an NV.

   ($double, $si) = Rmpz_get_d_2exp($op);
    Find $double and $si such that $double times 2 raised to
    $si, with 0.5<=abs($double)<1, is a good approximation to $op.

   $ul = Rmpz_getlimbn($op, $ui);
    Return limb number $ui from $op.  The sign of $op is ignored,
    just the absolute value is used.  The least significant limb
    is number 0. `Rmpz_size' can be used to find how many limbs
    make up $op. `Rmpz_getlimbn' returns zero if $ui is outside the
    range 0 to `Rmpz_size($op)-1'.

   $str = Rmpz_get_str($op, $base);
    Convert $op to a string of digits in base $base.
    The base may vary from -36..-2, 2..62.

   ##################

   INTEGER ARITHMETIC

   Rmpz_add($rop, $op1, $op2);
   Rmpz_add_ui($rop, $op, $ui);
    $rop = 2nd arg + 3rd arg.

   Rmpz_sub($rop, $op1, $op2);
   Rmpz_sub_ui($rop, $op, $ui);
   Rmpz_ui_sub($rop, $ui, $op);
    $rop = 2nd arg - 3rd arg.

   Rmpz_mul($rop, $op1, $op2);
   Rmpz_mul_si($rop, $op, $si);
   Rmpz_mul_ui($rop, $op, $ui);
    $rop = 2nd arg * 3rd arg.

   Rmpz_addmul($rop, $op1, $op2);
   Rmpz_addmul_ui($rop, $op, $ui);
    $rop += 2nd arg * 3rd arg.

   Rmpz_submul($rop, $op1, $op2);
   Rmpz_submul_ui($rop, $op, $ui);
    $rop -= 2nd arg * 3rd arg.

   Rmpz_mul_2exp($rop, $op, $ui);
    Set $rop to $op * (2 ** $ui).  This operation can
    also be defined as a left shift by $ui bits.

   Rmpz_div_2exp($rop, $op, $ui); # Same as Rmpz_fdiv_q_2exp
    Set $rop to $op / (2 ** $ui).  This operation can
    also be defined as a right shift by $ui bits.

   Rmpz_neg($rop, $op);
    $rop = -$op.

   Rmpz_abs($rop, $op);
    $rop = abs($op).

   ################

   INTEGER DIVISION

   `cdiv' rounds quotient up towards +infinity, and remainder
          will have the opposite sign to divisor.
         The `c' stands for "ceil".

   `fdiv' rounds quotient down towards -infinity, and remainder
          will have the same sign as divisor.
          The `f' stands for "floor".

   `tdiv' rounds quotient towards zero, and remainder
          will have the same sign as the number.
          The `t' stands for "truncate".

   Rmpz_div($rop, $op1, $op2); # Same as Rmpz_fdiv_q
   Rmpz_cdiv_q($rop, $op1, $op2);
   Rmpz_fdiv_q($rop, $op1, $op2);
   Rmpz_tdiv_q($rop, $op1, $op2);
    $rop = $op1 / $op2.

   Rmpz_cdiv_r($rop, $op1, $op2);
   Rmpz_fdiv_r($rop, $op1, $op2);
   Rmpz_tdiv_r($rop, $op1, $op2);
    $rop = $op1 % $op2.

   Rmpz_divmod($rop1, $rop2m $op1, $op2); # Same as Rmpz_fdiv_qr
   Rmpz_cdiv_qr($rop1, $rop2, $op1, $op2);
   Rmpz_fdiv_qr($rop1, $rop2, $op1, $op2);
   Rmpz_tdiv_qr($rop1, $rop2, $op1, $op2);
    $rop1 = $op1 / $op2.
    $rop2 = $op1 % $op2.

   $ul = Rmpz_div_ui($rop, $op, $ui); # Same as Rmpz_fdiv_q_ui
   $ul = Rmpz_cdiv_q_ui($rop, $op, $ui);
   $ul = Rmpz_fdiv_q_ui($rop, $op, $ui);
   $ul = Rmpz_tdiv_q_ui($rop, $op, $ui);
    $rop = $op / $ui.
    $ul = $op % $ui.

   $ul = Rmpz_cdiv_r_ui($rop, $op, $ui);
   $ul = Rmpz_fdiv_r_ui($rop $op, $ui);
   $ul = Rmpz_tdiv_r_ui($rop, $op, $ui);
    $rop = $op % $ui.
    $ul = $op % $ui.

   $ul = Rmpz_divmod_ui($rop1, $rop2, $op1, $ui); # Same as Rmpz_fdiv_qr_ui
   $ul = Rmpz_cdiv_qr_ui($rop1, $rop2, $op, $ui);
   $ul = Rmpz_fdiv_qr_ui($rop1, $rop2, $op, $ui);
   $ul = Rmpz_tdiv_qr_ui($rop1, $rop2, $op, $ui);
    $rop1 = $op / $ui.
    $rop2 = $op % $ui.
    $ul = $op % $ui.

   $ul = Rmpz_cdiv_ui($op, $ui);
   $ul = Rmpz_fdiv_ui($op, $ui);
   $ul = Rmpz_tdiv_ui($op, $ui);
    $ul = $op % $ui.

   Rmpz_cdiv_q_2exp($rop, $op, $ui);
   Rmpz_fdiv_q_2exp($rop, $op, $ui);
   Rmpz_tdiv_q_2exp($rop, $op, $ui);
    $rop = $op / (2 ** $ui). ie $rop is $op right-shifted
    by $ui bits.

   Rmpz_mod_2exp($rop, $op, $ui); # Same as Rmpz_fdiv_r_2exp
   Rmpz_cdiv_r_2exp($rop, $op, $ui);
   Rmpz_fdiv_r_2exp($rop, $op, $ui);
   Rmpz_tdiv_r_2exp($rop, $op, $ui);
    $rop = $op % (2 ** $ui).

   Rmpz_mod($rop, $op1, $op2);
    $rop = $op1 % $op2. The sign of the divisor is ignored.
    The result is never negative.

   $ul = Rmpz_mod_ui($rop, $op, $ui);
    $rop = $op % $ui.
    $ul = $op % $ui.
    The sign of the divisor is ignored. The result is never negative.

   Rmpz_divexact($rop, $op1, $op2);
   Rmpz_divexact_ui($rop, $op, $ui);
    $rop = 2nd arg / 3rd arg.
    These 2 functions provide correct results only when the
    2nd arg is exactly divisible by the 3rd arg.
    (Exactness can be checked by using Rmpz_divisible_p and
    Rmpz_divisible_ui_p, below.)

   $bool = Rmpz_divisible_p($op1, $op2);
   $bool = Rmpz_divisible_ui_p($op, $ui);
   $bool = Rmpz_divisible_2exp_p($op, $ui);
    Return non-zero if 1st arg is exactly divisible by 2nd arg,
    or in the case of `Rmpz_divisible_2exp_p' by 2 ** 2nd arg.

   $bool = Rmpz_congruent_p($op1, $op2, $op3);
   $bool = Rmpz_congruent_ui_p($op, $ui, $ui);
   $bool = Rmpz_congruent_2exp_p($op1, $op2, $ui);
    Return non-zero if 1st arg is congruent to 2nd arg modulo
    3rd arg, or in the case of `Rmpz_congruent_2exp_p' modulo
    2 ** 3rd arg.

   ######################

   INTEGER EXPONENTIATION

   Rmpz_powm($rop, $op1, $op2, $op3);
   Rmpz_powm_sec($rop, $op1, $op2, $op3); # gmp-5.0 and later only
    $rop = ($op1 ** $op2 ) % $op3
    In the case of Rmpz_powm_sec, $op2 must be > 0, and $op3 must
    be odd.

   Rmpz_powm_ui($rop, $op1, $ui, $op2);
    $rop = ($op1 ** $ui) % $op2

   Rmpz_pow_ui($rop, $op, $ui);
    $rop = $op ** $ui

   Rmpz_ui_pow_ui($rop, $ui1, $ui2);
    $rop = $ui1 ** $ui2

   #############

   INTEGER ROOTS

   Rmpz_root($rop, $op, $ui);
    $rop = $op ** (1 / $ui).

   Rmpz_rootrem($rop1, $rop2, $op, $ui);
    $rop1 = $op **(1 / $ui).
    $op = $rop2 + ($rop1 ** $ui).

   Rmpz_sqrt($rop, $op);
    $rop = $op ** 0.5.

   Rmpz_sqrtrem($rop1, $rop2, $op);
    $rop1 = $op ** 0.5.
    $op = $rop2 + ($rop1 ** 2).

   $bool = Rmpz_perfect_power_p($op);
    Return zero if $op is not a perfect power.
    Else return non-zero.

   $bool = Rmpz_perfect_square_p($op);
    Return zero if $op is not a perfect square.
    Else return non-zero.

   ##########################

   NUMBER THEORETIC FUNCTIONS

   $si = Rmpz_probab_prime_p($rop, $ui);
    Determine whether $rop is prime. Return 2 if $rop is
    definitely prime, return 1 if $rop is probably prime
    (without being certain), or return 0 if $rop is definitely
    composite. This function does some trial divisions, then
    some Miller-Rabin probabilistic primality tests.  $ui
    controls how many such tests are done, 5 to 10 is a
    reasonable number, more will reduce the chances of a
    composite being returned as "probably prime".
    Miller-Rabin and similar tests can be more properly called
    compositeness tests.  Numbers which fail are known to be
    composite but those which pass might be prime or might be
    composite. Only a few composites pass, hence those which
    pass are considered probably prime.

   Rmpz_nextprime($rop, $op);
   $ui = Rmpz_prevprime($rop, $op);
    NEXT: Set $rop to the smallest prime > $op
    PREV: Set $rop to the biggest  prime < $op
    Rmpz_prevprime returns 1 if $rop is probably prime and 2 if
    $rop is definitely prime. If a previous prime doesn't exist
    (i.e. $op < 3), Rmpz_prevprime returns 0 without altering $rop.
    These functions use a probabilistic algorithm to identify
    primes. For practical purposes it's adequate; the chance
    of a composite passing will be extremely small.

   Rmpz_gcd($rop, $op1, $op2);
    Set $rop to the greatest common divisor of $op1 and $op2.
    The result is always positive even if one or both input
    operands are negative.

   $ui2 = Rmpz_gcd_ui($rop, $op, $ui1);  # GCD in $ui2 & $rop
   $ui2 = Rmpz_gcd_ui($NULL, $op, $ui1); # GCD in $ui2 only
    Compute the greatest common divisor of $op and $ui1.
    Store the result in $rop (iff $rop is not $Math::GMPz::NULL).
    If the result is small enough to fit in an `unsigned long
    int', it is returned.  If the result does not fit, 0 is
    returned, and the result is equal to the argument $op.
    Note that the result will always fit if $ui1 is non-zero.

   Rmpz_gcdext($rop1, $rop2, $rop3, $op1, $op2);
   Rmpz_gcdext($rop1, $rop2, $NULL, $op1, $op2);
    Set $rop1 to the greatest common divisor of $op1 and $op2,
    and in addition set $rop2 and $rop3 to coefficients
    satisfying $op1*$rop2 + $op2*$rop3 = $rop1. $rop1 is
    always positive, even if one or both of $op1 and $op2
    are negative.
    If $rop3 is $Math::GMPz::NULL then that value is not
    computed.

   Rmpz_lcm($rop, $op1, $op2);
   Rmpz_lcm_ui($rop, $op, $ui);
    Set $rop to the least common multiple of 2nd and 3rd args.
    $rop is always positive, irrespective of the signs of the
    2nd and 3rd args. $rop will be zero if either
    2nd or 3rd arg is zero.

   $bool = Rmpz_invert($rop, $op1, $op2);
    Compute the inverse of $op1 modulo $op2 and put the result
    in $rop. If the inverse exists, the return value is
    non-zero and $rop will satisfy 0 <= $rop < $op2.
    If an inverse doesn't exist the return value is zero and
    $rop is undefined.

   $si = Rmpz_jacobi($op1, $op2);
    Calculate the Jacobi symbol ($op1/$op2). This is defined
    only for $op2 odd.

   $si = Rmpz_legendre($op1, $op2);
    Calculate the Legendre symbol ($op1/$op2). This is defined
    only for $op2 an odd positive prime, and for such $op2
    it's identical to the Jacobi symbol.

   $si = Rmpz_kronecker($op1, $op2);
   $si = Rmpz_kronecker_si($op, $si);
   $si = Rmpz_kronecker_ui($op, $ui);
   $si = Rmpz_si_kronecker($si, $op);
   $si = Rmpz_ui_kronecker($ui, $op);
    Calculate the Jacobi symbol (1st arg/2nd arg) with the
    Kronecker extension (a/2)=(2/a) when a odd, or (a/2)=0
    when a even. When 2nd arg is odd the Jacobi symbol and
    Kronecker symbol are identical, so 'Rmpz_kronecker_ui'
    etc can be used for mixed precision Jacobi symbols too.

   $ui = Rmpz_remove($rop, $op1, $op2);
    Remove all occurrences of the factor $op2 from $op1 and
    store the result in $rop.  The return value is how many
    such occurrences were removed.

   Rmpz_fac_ui($rop, $ui);
    Set $rop to the factorial of $ui.

   Rmpz_2fac_ui($rop, $ui); # Available only with gmp-5.1.0
                            # or later
    Set $rop to the double-factorial (n!!) of $ui.

   Rmpz_mfac_uiui($rop, $ui1, $u2); # Available only with
                                    # gmp-5.1.0 or later
    Set $rop to the $ui2-multi-factorial of $ui1, $ui2.

   Rmpz_primorial_ui($rop, $ui); # Available only with gmp-5.1.0
                                 # or later
    Set $rop to the primorial of $ui, i.e. the product of all
    positive prime numbers smaller than or equal to $ui.

   Rmpz_bin_ui($rop, $op, $ui);
   Rmpz_bin_uiui($rop, $ui, $ui);
   Rmpz_bin_si($rop, $op, $si);
    Compute the binomial coefficient 2nd arg over 3rd arg and
    store the result in $rop.  Negative values of 2nd arg are
    supported by `Rmpz_bin_ui' & `Rmpz_bin_si'.
    (NOTE: The GMP library does not provide an `mpz_bin_si'
    function.)

   Rmpz_fib_ui($rop, $ui);
   Rmpz_fib2_ui($rop1, $rop2, $ui);
    `Rmpz_fib_ui' sets $rop to to F[$ui], the $ui'th
    Fibonacci number. `Rmpz_fib2_ui' sets $rop1 to F[$ui],
    and $rop2 to F[$ui-1]. These functions are designed for
    calculating isolated Fibonacci numbers.  When a sequence
    of values is wanted it's best to start with `Rmpz_fib2_ui'
    and iterate the defining F[n+1]=F[n]+F[n-1] or similar.

   Rmpz_lucnum_ui($rop, $ui);
   Rmpz_lucnum2_ui($rop1, $rop2, $ui);
    `Rmpz_lucnum_ui' sets $rop to to L[$ui], the $ui'th
    Lucas number. `Rmpz_lucnum2_ui' sets $rop1 to L[$ui],
    and $rop2 to L[$ui-1]. These functions are designed for
    calculating isolated Lucas numbers.  When a sequence of
    values is wanted it's best to start with `Rmpz_lucnum2_ui'
    and iterate the defining L[n+1]=L[n]+L[n-1] or similar.

   ###################

   INTEGER COMPARISONS

   $si = Rmpz_cmp($op1, $op2);
   $si = Rmpz_cmp_d($op, $double);
   $si = Rmpz_cmp_NV($op, $nv); # $nv is an NV
   $si = Rmpz_cmp_IV($op, $iv); # $iv is an IV
   $si = Rmpz_cmp_si($op, $si);
   $si = Rmpz_cmp_ui($op, $ui);
   $si = Rmpz_cmp_sj($op, $si); # Available if longsize < ivsize
   $si = Rmpz_cmp_uj($op, $ui); # Available if longsize < ivsize
    Compare 1st and 2nd args.  Return a positive value if
    1st arg > 2nd arg, zero if 1st arg = 2nd arg, or a
    negative value if 1st arg < 2nd arg.
    Note that Rmpz_cmp_d and Rmpz_cmp_NV compare $op to the
    actual value held in the double/NV - not to the truncated
    integer value. Comparison to an Inf is allowed, but
    comparison to a NaN is a fatal error.
    Rmpz_cmp_sj and Rmpz_cmp_uj do not have gmp library counterparts
    and are available only on perls where ivsize > longsize.

    NOTE:
         Rmpz_cmp_IV() requires that the 2nd argument has its IOK
         flag set, and Rmpz_cmp_NV() requires that the 2nd
         argument has its NOK flag set.
         Otherwise these functions croak.
         Suggestion: first check the status of the flag using
         IOK_flag($iv) or NOK_flag($nv),which return a non-zero
         value if and only if the flag in question is set.

   $si = Rmpz_cmpabs($op1, $op2);
   $si = Rmpz_cmpabs_d($op, $double);
   $si = Rmpz_cmpabs_ui($op, $ui);
    Compare the absolute values of 1st and 2nd args.  Return
    a positive value if abs(1st arg) > abs(2nd arg), zero if
    abs(1st arg) = abs(2nd arg), or a negative value if
    abs(1st arg) < abs(2nd arg).

   $si = Rmpz_sgn($op);
    Return +1 if $op > 0, 0 if $op == 0, and -1 if $op < 0.

   ##############################

   INTEGER LOGIC AND BIT FIDDLING

   Rmpz_and($rop, $op1, $op2);
    Set $rop to $op1 logical-and $op2.

   Rmpz_ior($rop, $op1, $op2);
    Set $rop to $op1 inclusive-or $op2.

   Rmpz_xor($rop, $op1, $op2);
    Set $rop to $op1 exclusive-or $op2.

   Rmpz_com($rop, $op);
    Set $rop to the one's complement of $op.

   $ui = Rmpz_popcount($op);
    (This function returns an unsigned long value. If the actual
    popcount is greater than the maximum value that an unsigned
    long can hold, then the return value will be incorrect.)
    If $op>=0, return the population count of $op, which is the
    number of 1 bits in the binary representation.  If $op<0, the
    number of 1s is infinite, and the return value is MAX_ULONG,
    the largest possible `unsigned long'.

   $ui = Rmpz_hamdist($op1, $op2);
    (This function returns an unsigned long value. If the actual
    hamming distance is greater than the maximum value that an
    unsigned long can hold, then the return value will be
    incorrect.)
    If $op1 and $op2 are both >=0 or both <0, return the hamming
    distance between the two operands, which is the number of bit
    positions where $op1 and $op2 have different bit values.  If
    one operand is >=0 and the other <0 then the number of bits
    different is infinite, and the return value is MAX_ULONG,
    the largest possible `unsigned long'.

   $ui = Rmpz_scan0($op, $ui);
   $ui = Rmpz_scan1($op, $ui);
    Scan $op, starting from bit index $ui, towards more
    significant bits, until the first 0 or 1 bit (respectively)
    is found.  Return the index of the found bit. If the bit at
    index $ui is already what's sought, then $ui is returned.
    If there's no bit found, then MAX_ULONG is returned.  This
    will happen in `Rmpz_scan0' past the end of a positive
    number, or `Rmpz_scan1' past the end of a negative.

   Rmpz_setbit($rop, $ui);
    Set bit index $ui in $rop.

   Rmpz_clrbit($rop, $ui);
    Clear bit index $ui in $rop.

   $si = Rmpz_tstbit($op, $ui);
    Test bit index $ui in $op and return 0 or 1 accordingly.

   ###############

   I/O of INTEGERS

   $bytes_written = Rmpz_out_str([$prefix,] $op, $base [, $suffix]);

    BEST TO USE TRmpz_out_str INSTEAD.
    Output $op to STDOUT, as a string of digits in base $base.
    The base may vary from -36..-2, 2..62. Return the number
    of bytes written, or if an error occurred,return 0.
    The optional arguments ($prefix and $suffix) are strings
    that will be prepended/appended to the mpz_out_str output.
    $bytes_written does not include the bytes contained in
    $prefix and $suffix.

   $bytes_written = TRmpz_out_str([$prefix,] $stream, $base, $op, [, $suffix]);
    (This function doesn't work with unblessed objects.)
    As for Rmpz_out_str, except that there's the capability to print
    to somewhere other than STDOUT. Note that the order of the args
    is different (to match the order of the mpz_out_str args).
    To print to STDERR:
       TRmpz_out_str(*stderr, $base, $digits, $op);
    To print to an open filehandle (let's call it $fh):
       TRmpz_out_str(\*$fh, $base, $digits, $op);

   $bytes_written = Rmpz_out_raw(\*$fh, $op);
    Output $op to filehandle $fh, in raw binary format. The integer is
    written in a portable format, with 4 bytes of size information, and
    that many bytes of limbs. Both the size and the limbs are written
    in decreasing significance order (i.e., in big-endian). The output
    can be read with mpz_inp_raw.

   $bytes_read = Rmpz_inp_str($rop, $base);
    BEST TO USE TRmpz_inp_str instead.
    Input a string in base $base from STDIN, and put the read
    integer in $rop. The base may vary from 2 to 62.  If $base
    is 0, the actual base is determined from the leading
    characters: if the first two characters are `0x' or `0X',
    hexadecimal is assumed, otherwise if the first character is
   `0', octal is assumed, otherwise decimal is assumed.
    Return the number of bytes read, or if an error occurred, return 0.

   $bytes_read = TRmpz_inp_str($rop, $stream, $base);
    (This function doesn't work with unblessed objects.)
    As for Rmpz_inp_str, except that there's the capability to read
    from somewhere other than STDIN.
    To read from STDIN:
       TRmpz_inp_str($rop, *stdin, $base);
    To read from an open filehandle (let's call it $fh):
       TRmpz_inp_str($rop, \*$fh, $base);

    $bytes_read = Rmpz_inp_raw($rop, \*$fh);
     Input from filehandle $fh in the format written by Rmpz_out_raw,
     and put the result in $rop. Return the number of bytes read, or
     if an error occurred, return 0.

   #######################

   RANDOM NUMBER FUNCTIONS

   In the random number functions, @r is an array of
   Math::GMPz objects (one for each random number that is
   required). $how_many is the number of random numbers you
   want and must be equal to scalar(@r). $bits is simply the
   number of random bits required. Before calling the random
   number functions, $state must be initialised and seeded.

   $state = rand_init($op); # $op is the seed.
    Initialises and seeds $state, ready for use with the random
    number functions. However, $state has not been blessed into
    any package, and therefore does not get cleaned up when it
    goes out of scope. To avoid memory leaks you therefore need
    to call 'rand_clear($state);' once you have finished with it
    and before it goes out of scope. Also, it uses the default
    algorithm. Consider using the following initialisation and
    seeding routines - they provide a choice of algorithm, and
    there's no need to call rand_clear() when you've finished with
    them.

   $state = zgmp_randinit_default();
    This is the Math::GMPz interface to the gmp library function
   'gmp_randinit_default'.
    $state is blessed into package Math::GMPz::Random and will be
    automatically cleaned up when it goes out of scope.
    Initialize $state with a default algorithm. This will be a
    compromise between speed and randomness, and is recommended for
    applications with no special requirements. Currently this is
    the gmp_randinit_mt function (Mersenne Twister algorithm).

   $state = zgmp_randinit_mt();
    This is the Math::GMPz interface to the gmp library function
   'gmp_randinit_mt'.
    Currently identical to zgmp_randinit_default().

   $state = zgmp_randinit_lc_2exp($mpz, $ui, $m2exp);
    This is the Math::GMPz interface to the gmp library function
   'gmp_randinit_lc_2exp'.
    $state is blessed into package Math::GMPz::Random and will be
    automatically cleaned up when it goes out of scope.
    Initialize $state with a linear congruential algorithm
    X = ($mpz*X + $ui) mod (2 ** $m2exp). The low bits of X in this
    algorithm are not very random. The least significant bit will have a
    period no more than 2, and the second bit no more than 4, etc. For
    this reason only the high half of each X is actually used.
    When a random number of more than m2exp/2 bits is to be generated,
    multiple iterations of the recurrence are used and the results
    concatenated.

   $state = zgmp_randinit_lc_2exp_size($ui);
    This is the Math::GMPz interface to the gmp library function
   'gmp_randinit_lc_2exp_size'.
    $state is blessed into package Math::GMPz::Random and will be
    automatically cleaned up when it goes out of scope.
    Initialize state for a linear congruential algorithm as per
    gmp_randinit_lc_2exp. a, c and m2exp are selected from a table,
    chosen so that $ui bits (or more) of each X will be used,
    ie. m2exp/2 >= $ui.
    If $ui is bigger than the table data provides then the function fails
    and dies with an appropriate error message. The maximum value for $ui
    currently supported is 128.

   $state2 = zgmp_randinit_set($state1);
    This is the Math::GMPz interface to the gmp library function
   'gmp_randinit_set'.
    $state2 is blessed into package Math::GMPz::Random and will be
    automatically cleaned up when it goes out of scope.
    Initialize $state2 with a copy of the algorithm and state from
    $state1.

   $state = zgmp_randinit_default_nobless();
   $state = zgmp_randinit_mt_nobless();
   $state = zgmp_randinit_lc_2exp_nobless($mpz, $ui, $m2exp);
   $state2 = zgmp_randinit_set_nobless($state1);
    As for the above comparable function, but $state is not blessed into
    any package. (Generally not useful - but they're available if you
    want them.)

   zgmp_randseed($state, $mpz);
   zgmp_randseed_ui($state, $ui);
    These are the Math::GMPz interfaces to the gmp library functions
   'gmp_randseed' and 'gmp_randseed_ui'.
    Seed an initialised (but not yet seeded) $state with $mpz/$ui.

   $ui = zgmp_urandomb_ui($state, $bits);
    This is the Math::GMPz interface to the gmp library function
    'gmp_urandomb_ui'.
    Return a uniformly distributed random number of $bits bits, ie. in
    the range 0 to 2 ** ($bits - 1) inclusive. $bits must be less than or
    equal to the number of bits in an unsigned long.

   $ui2 = zgmp_urandomm_ui($state, $ui1);
    This is the Math::GMPz interface to the gmp library function
    'gmp_urandomm_ui'.
    Return a uniformly distributed random number in the range 0 to
    $ui1 - 1, inclusive.

   Rmpz_urandomm(@r, $state, $mpz, $how_many);
    Generate $how_many uniform random integers in the range
    0 to $op-1, inclusive.

   Rmpz_urandomb(@r, $state, $bits, $how_many);
    Generate $how_many uniformly distributed random integers
    in the range 0 to 2**($bits-1), inclusive.

   Rmpz_rrandomb(@r, $state, $bits, $how_many);
    Generate $how_many random integers with long strings of
    zeros and ones in the binary representation. Useful for
    testing functions and algorithms, since this kind of random
    numbers have proven to be more likely to trigger corner-case bugs.
    The random number will be in the range 0 to 2**($bits-1), inclusive.

   zgmp_randclear($state);
   rand_clear($state);
    Destroys $state, as also does Math::GMPz::Random::DESTROY - three
    identical functions.
    Use only if $state is an unblessed object - ie if it was initialised
    using rand_init() or one of the zgmp_randinit*_nobless functions.

   #########################

   INTEGER IMPORT AND EXPORT

    NOTE: There's a demo (demos/imp_exp.p) in the source distro. It
          contains comments that will hopefully explain the basics of the
          import and export functions, if it's not clear from the following
          documentation.

   Rmpz_import($rop, $len, $order, $size, $endian, $nails, $str);

    Take a string ($str) and convert it to a Math::GMPz object ($rop)
    treating the string as a base 256 number.
    $len many words are read, each of $size bytes.
    Normally, $len = length($str) with $size set to 1,
    but you can opt not to take the entire string if you like.
    Yet another alternative is to set $len = 1, and
    $size = length($str).

    $order can be 1 for most significant byte first or -1 for least significant
    byte first.

    $endian can be 1 for most significant byte within each word first, -1 for
    least significant byte in each word first, or 0 for native endianness.
    If $size is 1, then the value given to $endian is irrelevant.

    The most significant $nails bits of each word are skipped.

    UTF8 strings call for special treatment.
    Rmpz_import, by default, does a utf8::downgrade of such strings before
    calculating the value to be assigned to $rop.
    Before terminating, Rmpz_import will then perform a utf8::upgrade of $str
    if and only if $str was initially UTF8.

    This automatic utf8::downgrade can be disallowed by setting the package
    global $Math::GMPz::utf8_no_downgrade to 1.

    If the automatic downgrade is permitted ($Math::GMPz::utf8_no_downgrade == 0,
    which is the default), then a warning will be emitted every time Rmpz_import
    receives a UTF8 string.
    This warning can be suppressed by setting $Math::GMPz::utf8_no_warn to 1.
    The default value of $Math::GMPz::utf8_no_warn = 0 enables the warning.

    If the utf8::downgrade fails, default behaviour ($Math::GMPz::utf8_no_croak = 0)
    is for the function to croak with the appropriate error message.
    If you wish to avoid the croak(), and have Rmpz_import continue on regardless,
    set $Math::GMPz::utf8_no_croak to 1.

    If $Math::GMPz::utf8_no_croak is set to a true value, and the utf8::downgrade
    fails then default behaviour ($Math::GMPz::utf8_no_fail = 0) is for you to be
    warned of that failure.
    To suppress such warnings, set $Math::GMPz::utf8_no_fail to 1.

    IOW, if you're passing UTF8 strings to Rmpz_import, it's probably best if you
    can do the downgrade and subsequent upgrade yourself - eg, something like:

      ...
      if(utf8::is_utf8($str)) {
        utf8::downgrade($str);
        Rmpz_import($rop, $len, $order, $size, $endian, $nails, $str);
        utf8::upgrade($str)
      }
      else {
        Rmpz_import($rop, $len, $order, $size, $endian, $nails, $str);
      }
      ...

    That way, you don't have to concern yourself with those 4 package global
    scalars at all !!

    You might (or might not) find t/imp_exp.t from the Math-GMPz test suite
    helpful in comprehending this documentation.

   $str = Rmpz_export($order, $size, $endian, $nails, $op);

    Returns a base 256 ASCII string representation of the number held by
    the Math::GMPz object, "$op".  While Rmpz_import converts a string to a
    number, this function can convert that number to an ASCII string.
    Assuming that Rmpz_export and Rmpz_import receive a value of $nails = 0
    && that they both also receive the same values for $order, $size and
    $endian, then this returned $str will be equal to the original string
    passed to Rmpz_import if and only if:
     a) that original string was not UTF8;
       or
     b) the original string was UTF8 && was successfully downgraded during
        the Rmpz_import processing.

    However, in the case of b), the fact that the original string was UTF8
    suggests that you might want to utf8::upgrade the string returned by
    Rmpz_export.

   Rmpz_import_UV($rop, $len, $order, $size, $endian, $nails, $arref);

    As for Rmpz_import, except that the final argument is a reference to an
    array of UVs (unsigned perl integers).

   $arref = Rmpz_export_UV($order, $size, $endian, $nails, $op);

    As for Rmpz_export, except that it returns a reference to an array of
    UVs (unsigned perl integers).
    Assuming that Rmpz_export_UV and Rmpz_import_UV receive a value of
    $nails = 0 && that they both also receive the same values for $order,
    $size and $endian, then the array of UVs returned by this function, when
    given to Rmpz_import_UV will generate a Math::GMPz object whose value is
    equivalent to that of $op.

   ###############################

   MISCELLANEOUS INTEGER FUNCTIONS

   $bool = Rmpz_fits_ulong_p ($op);
   $bool = Rmpz_fits_slong_p ($op);
   $bool = Rmpz_fits_uint_p  ($op);
   $bool = Rmpz_fits_sint_p  ($op);
   $bool = Rmpz_fits_ushort_p($op);
   $bool = Rmpz_fits_sshort_p($op);
   $bool = Rmpz_fits_IV_p    ($op);

    Return non-zero iff the value of $op fits an 'unsigned long int',
    'signed long int', 'unsigned int', 'signed int', 'unsigned short
    int', or 'signed short int', or 'IV' respectively.
    Otherwise, return zero.

   $si = IOK_flag($sv); # $sv is a perl scalar variable.
   $si = NOK_flag($sv);
   $si = POK_flag($sv);

   Return 0 if $sv's IOK/NOK/POK flag is unset.
   Else return 1.
   If the IsUV flag is set, then IOK_flag() returns 2, thereby indicating
   that both the IOK and IsUV flags are set (and that the integer value
   held by $sv should therefore be treated as unsigned).

   $bool = Rmpz_odd_p ($op);
   $bool = Rmpz_even_p($op);
    Determine whether $op is odd or even, respectively.
    Return non-zero if yes, zero if no.

   $ui = Rmpz_size($op);
    Return the size of $op measured in number of limbs.
    If $op is zero, the returned value will be zero.

   $ui = Rmpz_sizeinbase($op, $base);
    (This function returns a size_t value. If the actual sizeinbase
    of $op is greater than the maximum value that a size_t can hold,
    then the return value will be incorrect.)
    Return the size of $op measured in number of digits in base
    $base. The base may vary from 2 to 62.  The sign of $op is
    ignored, just the absolute value is used.  The result will be
    exact or 1 too big.  If $base is a power of 2, the result will
    always be exact. If $op is zero the return value is always 1.

   ####################

   OPERATOR OVERLOADING

    Overloading works with numbers, strings (integer only), Math::GMPz
    objects, and Math::BigInt objects. Non-integer NV values will be
    truncated (ie rounded towards zero) to an integer value.

    I believe (not rigorously tested) that wherever the Math::GMPz
    operator overloading accepts a Math::GMPz object as the second arg,
    it will alternatively accept (and understand) a Math::GMP object as
    that second arg. Indications are that Math::GMP reciprocates.
    See also "INTEROPERABILITY WITH GMP", below.

    YMMV if you use old versions of Math::GMPq and Math::MPFR, but if you
    have Math-GMPq-0.61 (or later) and Math-MPFR-4.37 (or later) then, apart
    from the exceptions noted below, you will be able to use Math::GMPq and
    Math::MPFR objects with the overloaded Math::GMPz operations. Such
    operations return Math::GMPq and (respectively) Math::MPFR objects.

   The following operators are overloaded:
    + - * / % **         # '**'  not usable with Math::GMPq objects.
                         # Returned Math::MPFR objects will have been
                         # calculated with default precision and rounding.

    += -= *= /= %= **=   # '**=' not usable with Math::GMPq objects.
                         # Returned Math::MPFR objects will have been
                         # calculated to the same precision as the given
                         # Math::MPFR operand, and with default rounding.

    << >> <<= >>=   # Not usable with Math::GMPq and Math::MPFR objects.

    & | ^ ~         # Not usable with Math::GMPq and Math::MPFR objects
    &= |= ^=        # Not usable with Math::GMPq and Math::MPFR objects

    < <= > >= == != <=>
    ! abs
    =

    NOTE: Making use of the '=' overloading is not recommended unless
          you understand its caveats. See 'perldoc overload' and
          read it thoroughly, including the documentation regarding
          'copy constructors'.

    Division uses 'tdiv' (see 'Integer Division', above).
    Check that '~', '%', and '%=' are working as you expect
    (especially in relation to negative values).

    Attempting to use the overloaded operators with objects that have
    been blessed into some package other than 'Math::BigInt', 'Math::GMP',
   'Math::GMPz', 'Math::GMPq', or 'Math::MPFR' will not work.

    To determine the type and value of the second argument the subroutine
    works through the following steps (in order), using the first value
    it  finds, or croaking if it gets to step 6:

    1. If the variable is a UV then that value is used. The variable
       is considered to be a UV if the IOK and IsUV flags are set.

    2. If the variable is an IV, then that value is used.
       The variable is considered to be an IV if the IOK flag is set.

    3. If the variable is a string (ie the POK flag is set) then the
       value of that string is used. Octal strings must begin with
       '0', hex strings must begin with either '0x' or '0X' -
       otherwise the string is assumed to be decimal. If the POK
       flag is set, but the string is not a valid base 8, 10, or 16
       number, the subroutine croaks with an appropriate error
       message.

    4. If the variable is an NV, then that value is truncated to an
       integer value and used. The variable is  considered to be an
       NV if the NOK flag is set.

    5. If the variable is a Math::GMPz object (or, for operators
       specified above, a Math::MPFR/Math::GMP/Math::GMPq/Math::BigInt
       object) then the value of that object is used.

    6. If none of the above is true, then the second variable is
       deemed to be of an invalid type. The subroutine croaks with
       an appropriate error message.

   #####

   OTHER

   $GMP_version = Math::GMPz::gmp_v;
    Returns the version of the GMP library (eg 4.2.3) being used by
    Math::GMPz. The function is not exportable.

   $GMP_cc = Math::GMPz::__GMP_CC;
   $GMP_cflags = Math::GMPz::__GMP_CFLAGS;
    If Math::GMPz has been built against gmp-4.2.3 or later,
    returns respectively the CC and CFLAGS settings that were used
    to compile the gmp library against which Math::GMPz was built.
    (Values are as specified in the gmp.h that was used to build
    Math::GMPz.)
    Returns undef if Math::GMPz has been built against an earlier
    version of the gmp library.
    (These functions are in @EXPORT_OK and are therefore exportable
    by request. They are not listed under the ":mpz" tag.)

   $major = Math::GMPz::__GNU_MP_VERSION;
   $minor = Math::GMPz::__GNU_MP_VERSION_MINOR;
   $patchlevel = Math::GMPz::__GNU_MP_VERSION_PATCHLEVEL;
    Returns respectively the major, minor, and patchlevel numbers
    for the GMP library version used to build Math::GMPz. Values are
    as specified in the gmp.h that was used to build Math::GMPz.
    (These functions are in @EXPORT_OK and are therefore exportable
    by request. They are not listed under the ":mpz" tag.)

   ################

   FORMATTED OUTPUT

   NOTE: The format specification can be found at:
   http://gmplib.org/manual/Formatted-Output-Strings.html#Formatted-Output-Strings
   However, the use of '*' to take an extra variable for width and
   precision is not allowed in this implementation. Instead, it is
   necessary to interpolate the variable into the format string - ie,
   instead of:
     Rmpz_printf("%*Zd\n", $width, $mpz);
   we need:
     Rmpz_printf("%${width}Zd\n", $mpz);

   $si = Rmpz_printf($format_string, $var);

    This function changed with the release of Math-GMPz-0.27.
    Now (unlike the GMP counterpart), it is limited to taking 2
    arguments - the format string, and the variable to be formatted.
    That is, you can format only one variable at a time.
    If there is no variable to be formatted, then the final arg
    can be omitted - a suitable dummy arg will be passed to the XS
    code for you. ie the following will work:
     Rmpz_printf("hello world\n");
    Returns the number of characters written, or -1 if an error
    occurred.

   $si = Rmpz_fprintf($fh, $format_string, $var);

    This function (unlike the GMP counterpart) is limited to taking
    3 arguments - the filehandle, the format string, and the variable
    to be formatted. That is, you can format only one variable at a time.
    If there is no variable to be formatted, then the final arg
    can be omitted - a suitable dummy arg will be passed to the XS
    code for you. ie the following will work:
     Rmpz_fprintf($fh, "hello world\n");
    Returns the number of characters written, or -1 if an error
    occurred.

   $si = Rmpz_sprintf($buffer, $format_string, [$var,] $buflen);

    This function (unlike the GMP counterpart) is limited to taking
    4 arguments - the buffer, the format string, the variable to be
    formatted and the buffer length. If there is no variable to be
    formatted, then the "$var" arg can be omitted - a suitable dummy
    arg will be passed to the XS code for you. ie the following will
    work:
     Rmpz_sprintf($buffer, "hello world\n", 12);
    $buflen must be large enough to accommodate the formatted string,
    Sets $buffer to the formatted string and returns the number of
    characters written, or -1 if an error occurred.

   $si = Rmpz_snprintf($buffer, $bytes, $format_string, [$var,] $buflen);

    Form a null-terminated string in $buffer. No more than $bytes
    bytes will be written. To get the full output, $bytes must be
    enough for the string and null-terminator. $buflen must be large
    enough to accommodate the string and null-terminator..
    The return value is the total number of characters which ought
    to have been produced, excluding the terminating null.
    If $si >= $bytes then the actual output has been truncated to
    the first $bytes-1 characters, and a null appended.
    This function (unlike the GMP counterpart) is limited to taking
    5 arguments - the buffer, the maximum number of bytes to be
    returned, the format string, the variable to be formatted, and
    the length of the buffer to which the formatted string will be
    written.
    If there is no variable to be formatted, then the "$var" arg can
    be omitted - a suitable dummy arg will be passed to the XS code
    for you. ie the following will work:
     Rmpz_snprintf($buffer, 6 "hello world", 12);

   ###############################

   INTEROPERABILITY WITH MATH::GMP

    In addition to Math::GMPz's overloaded operators accepting both
    Math::GMP and Math::GMPz objects (see "OPERATOR OVERLOADING"),
    Math::GMPz function calls seem to also accept Math::GMP and
    Math::GMPz objects interchangeably. (Not rigorously tested.)

   ###############################
   ###############################

BUGS
    You can get segfaults if you pass the wrong type of
    argument to the functions - so if you get a segfault, the
    first thing to do is to check that the argument types
    you have supplied are appropriate.

LICENSE
    This program is free software; you may redistribute it and/or
    modify it under the same terms as Perl itself.
    Copyright 2006-20011, 2013-24, Sisyphus

AUTHOR
    Sisyphus <sisyphus at(@) cpan dot (.) org>