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# NAME

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

# DEPENDENCIES

``````   This module needs the GMP C library - available from:
http://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 http://gmplib.org.
See the Math::GMPz test suite for examples of
usage.``````

# SYNOPSIS

``````   use Math::GMPz qw(:mpz :primes :supp);

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_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.

Rmpz_set_IV(\$rop, \$IV); # \$IV is a perl integer value
Set \$rop to the value of \$IV. (If \$IV is an unsigned integer
value then that unsigned value is assigned.)

Rmpz_set_NV(\$rop, \$NV); # \$NV is a perl floating point value
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.

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::BigInt object, a
number (signed integer, unsigned integer, signed fraction or
unsigned fraction) or a string that represents a numeric value. 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 .

\$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_NV(\$NV); # \$NV is a perl floating point value
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 functions
'Rmpz_fits_IV_p' and 'Rmpz_fits_UV_p'.

\$double = Rmpz_get_d(\$op);
Place the value of \$op into a normal perl scalar.

(\$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

\$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.

\$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, \$rop1, \$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 it
is known that the 3rd arg divides the 2nd arg.

\$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);
This function uses 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 `mpz_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
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
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_si(\$op, \$si);
\$si = Rmpz_cmp_ui(\$op, \$ui);
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.

\$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 \$opP = 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]);
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]);
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

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.

As for Rmpz_inp_str, except that there's the capability to read
from somewhere other than STDIN.
TRmpz_inp_str(\$rop, *stdin, \$base);
To read from an open filehandle (let's call it FH):
TRmpz_inp_str(\$rop, \*FH, \$base);

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

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

Take a binary string ("\$bstr") and convert it to a GMP
bignum structure, treating the string as a base 256
number. "\$rop" is a Math::GMPz object holding that number.
"\$len" is the length of the string segment to be
converted to the GMP bignum. Normally, \$len = length(\$bstr),
but you can opt not to take the entire string if you like.

Usually (\$order, \$size, \$endian, \$nails) = (1, 1, 0, 0);

See the GMP manual for a full explanation of what these
variables mean.

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

Rmpz_export() is simply the reverse of Rmpz_import().
It returns a base 256 string representation of the
number held by the Math::GMPz object, "\$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_UV_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', 'UV', or 'IV' respectively.
Otherwise, return zero.

\$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.

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

Math::BigInt objects and, to a limited extent, Math::GMPq objects
(iff the gmp library is version 6.1.0 or later) and Math::MPFR
objects (iff version 3.13 or later of Math::MPFR has been
installed).

+ - * / % **
+= -= *= /= %= **=
<< >> <<= >>=
& | ^ ~
&= |= ^=
< <= > >= == != <=>
! abs

The "power" operators ('**' and '**=') overload accept only +ve
integers (ie +IV or UV). Values that are too large to fit into
a C 'unsigned long int' will be silently truncated.

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::GMPz', 'Math::GMPq' (limited operations), or 'Math::MPFR'
(limited operations) will not work.
Math::MPFR objects can be used only with '+', '-', '*', '/'
and '**' operators, and will work only if Math::MPFR is at
version 3.13 or later - in which case the operation will
return a Math::MPFR object.
Math::GMPq objects can be used only with the comparison operators
( == != < <= > >= <=> ) and also, iff Math-GMPq-0.43 or later is
installed, with '+', '-', '/', and '*' operators (in which case
a Math::GMPq object is returned).

In those situations where the overload subroutine operates on 2
perl variables, then obviously one of those perl variables is
a Math::GMPz object. To determine the value of the other variable
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 an unsigned long then that value is used.
The variable is considered to be an unsigned long if
(perl 5.8) the UOK flag is set or if (perl 5.6) SvIsUV()
returns true. (In the case of perls whose ivtype is long long,
the variable is treated as an unsigned long long int if the
UOK/SvISUV flag is set.)

2. If the variable is a signed long int, then that value is used.
The variable is considered to be a signed long int if the
IOK flag is set. (In the case of perls whose ivtype is long long,
the variable is treated as a signed long long int if the IOK flag
is set.)

3. If the variable is an NV, then that value is used. The variable
is considered to be an NV if the NOK flag is set && the POK flag
is unset.

4. 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.

5. If the variable is a Math::GMPz object (or, for operators
specified above, a Math::MPFR/Math::GMPq 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.

If the second operand is a 'double' (ie if the other
operand's NOK flag is set) then it is first truncated
to an integer value before the operation is performed.

For example:
my \$x = Rmpz_init_set_ui(112);
\$x *= 2.9;
print "\$x"; # prints 224

#####

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,
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);

###################
###################``````

# 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.``````

``````    This program is free software; you may redistribute it and/or
``    Sisyphus <sisyphus at(@) cpan dot (.) org>``