# fmpz.h – integers

By default, an `fmpz_t` is implemented as an array of `fmpz`’s of length one to allow passing by reference as one can do with GMP’s `mpz_t` type. The `fmpz_t` type is simply a single limb, though the user does not need to be aware of this except in one specific case outlined below.

In all respects, `fmpz_t`’s act precisely like GMP’s `mpz_t`’s, with automatic memory management, however, in the first place only one limb is used to implement them. Once an `fmpz_t` overflows a limb then a multiprecision integer is automatically allocated and instead of storing the actual integer data the `slong` which implements the type becomes an index into a FLINT wide array of `mpz_t`’s.

These internal implementation details are not important for the user to understand, except for three important things.

Firstly, `fmpz_t`’s will be more efficient than `mpz_t`’s for single limb operations, or more precisely for signed quantities whose absolute value does not exceed `FLINT_BITS - 2`` bits.

Secondly, for small integers that fit into ``FLINT_BITS - 2`` bits much less memory will be used than for an `mpz_t`. When very many `fmpz_t`’s are used, there can be important cache benefits on account of this.

Thirdly, it is important to understand how to deal with arrays of `fmpz_t`’s. As for `mpz_t`’s, there is an underlying type, an `fmpz`, which can be used to create the array, e.g.

    fmpz myarr[100];

Now recall that an `fmpz_t` is an array of length one of `fmpz`’s. Thus, a pointer to an `fmpz` can be used in place of an `fmpz_t`. For example, to find the sign of the third integer in our array we would write

    int sign = fmpz_sgn(myarr + 2);

The `fmpz` module provides routines for memory management, basic manipulation and basic arithmetic.

Unless otherwise specified, all functions in this section permit aliasing between their input arguments and between their input and output arguments.

## Simple example

The following example computes the square of the integer \(7\) and prints the result.

    #include "fmpz.h"

    int main()
    {
        fmpz_t x, y;
        fmpz_init(x);
        fmpz_init(y);
        fmpz_set_ui(x, 7);
        fmpz_mul(y, x, x);
        fmpz_print(x);
        flint_printf("^2 = ");
        fmpz_print(y);
        flint_printf("\n");
        fmpz_clear(x);
        fmpz_clear(y);
    }

    7^2 = 49

## Types, macros and constants

type fmpz

  The FLINT multi-precision integer type uses an inline representation for small integers, specifically when the absolute value is at most \(2^{62}-1\) (on 64-bit machines) or \(2^{30}-1\) (on 32-bit machines). It switches automatically to a GMP integer for larger values.

  An `fmpz` is implemented as an `slong`. When its second most significant bit is \(0\) the `fmpz` represents an ordinary `slong` integer whose absolute value is at most `FLINT_BITS - 2` bits.

  When the second most significant bit is \(1\) then the value represents a pointer (the pointer is shifted right \(2\) bits and the second most significant bit is set to \(1\). This relies on the fact that `malloc` always allocates memory blocks on a \(4\) or \(8\) byte boundary).

type fmpz_t

  An array of length 1 of `fmpz`’s. This is used to pass `fmpz`’s around by reference without fuss, similar to the way `mpz_t` works.

COEFF_MAX

  The largest (positive) value an `fmpz` can be if just an `slong`.

COEFF_MIN

  The smallest (negative) value an `fmpz` can be if just an `slong`.

fmpz PTR_TO_COEFF(mpz_ptr ptr)

  A macro to convert an `mpz_t` (or more generally any `mpz_ptr`) to an `fmpz` (shifts the pointer right by \(2\) and sets the second most significant bit).

mpz_ptr COEFF_TO_PTR(fmpz f)

  A macro to convert an `fmpz` which represents a pointer into an actual pointer to an `__mpz_struct` (i.e. to an `mpz_t`).

COEFF_IS_MPZ(f)

  A macro which returns \(1\) if \(f\) represents an `mpz_t`, otherwise \(0\) is returned.

MPZ_MIN_ALLOC

  A constant determining the minimum number of limbs the fmpz memory manager will allocate for each mpz. This constant is currently 2.

mpz_ptr _fmpz_new_mpz(void)

  Returns a pointer to an initialised mpz with at least `MPZ_MIN_ALLOC` limbs allocated. This is only used internally.

  Note: Requires that `gmp.h` has been included before any FLINT header is included.

void _fmpz_clear_mpz(fmpz f)

  Clears the mpz “pointed to” by the fmpz f. This is only used internally.

Note

As of FLINT 3.2.0, it is required that the mpz pointed to by f has at least `MPZ_MIN_ALLOC` limbs allocated when `_fmpz_clear_mpz()` is called. Note that GMP functions never reduce the number of allocated limbs, apart from a bug in `mpz_remove` which as of GMP 6.3.0 may reduce the number of allocated limbs in the output mpz. Hence, as long as `mpz_realloc` never is called with a second argument less than `MPZ_MIN_ALLOC` on an mpz received from `_fmpz_new_mpz()`, any other usage should be fine.

void _fmpz_cleanup_mpz_content()

  This function does nothing in the reentrant version of `fmpz`.

void _fmpz_cleanup()

  This function does nothing in the reentrant version of `fmpz`.

mpz_ptr _fmpz_promote(fmpz_t f)

  If \(f\) doesn’t represent an `mpz_t`, initialise one and associate it to \(f\).

  Note: Requires that `gmp.h` has been included before any FLINT header is included.

mpz_ptr _fmpz_promote_val(fmpz_t f)

  If \(f\) doesn’t represent an `mpz_t`, initialise one and associate it to \(f\), but preserve the value of \(f\).

  This function is for internal use. The resulting `fmpz` will be backed by an `mpz_t` that can be passed to GMP, but the `fmpz` will be in an inconsistent state with respect to the other FLINT `fmpz` functions such as `fmpz_is_zero`, etc.

  Note: Requires that `gmp.h` has been included before any FLINT header is included.

void _fmpz_demote(fmpz_t f)

  If \(f\) represents an `mpz_t` clear it and make \(f\) just represent an `slong`.

void _fmpz_demote_val(fmpz_t f)

  If \(f\) represents an `mpz_t` and its value will fit in an `slong`, preserve the value in \(f\) which we make to represent an `slong`, and clear the `mpz_t`.

int _fmpz_is_canonical(const fmpz_t f)

  Returns 1 if the internal representation of \(f\) is correctly normalised and demoted; 0 otherwise.

## Memory management

void fmpz_init(fmpz_t f)

  A small `fmpz_t` is initialised, i.e. just a `slong`. The value is set to zero.

void fmpz_init2(fmpz_t f, ulong limbs)

  Initialises the given `fmpz_t` to have space for the given number of limbs.

  If `limbs` is zero then a small `fmpz_t` is allocated, i.e. just a `slong`. The value is also set to zero. It is not necessary to call this function except to save time. A call to `fmpz_init` will do just fine.

void fmpz_clear(fmpz_t f)

  Clears the given `fmpz_t`, releasing any memory associated with it, either back to the stack or the OS, depending on whether the reentrant or non-reentrant version of FLINT is built.

void fmpz_init_set(fmpz_t f, const fmpz_t g)

void fmpz_init_set_ui(fmpz_t f, ulong g)

void fmpz_init_set_si(fmpz_t f, slong g)

  Initialises \(f\) and sets it to the value of \(g\).

## Random generation

For thread-safety, the randomisation methods take as one of their parameters an object of type `flint_rand_t`. Before calling any of the randomisation functions such an object first has to be initialised with a call to `flint_rand_init()`. When one is finished generating random numbers, one should call `flint_rand_clear()` to clean up.

void fmpz_randbits_unsigned(fmpz_t f, flint_rand_t state, flint_bitcnt_t bits)
void fmpz_randbits(fmpz_t f, flint_rand_t state, flint_bitcnt_t bits)

  Generates a random integer whose absolute value has precisely the given number of bits.

void fmpz_randtest_unsigned(fmpz_t f, flint_rand_t state, flint_bitcnt_t bits)
void fmpz_randtest(fmpz_t f, flint_rand_t state, flint_bitcnt_t bits)

  Generates a random (unsigned) integer whose absolute value has a number of bits which is random from \(0\) up to `bits` inclusive.

void fmpz_randtest_not_zero(fmpz_t f, flint_rand_t state, flint_bitcnt_t bits)

  As per `fmpz_randtest`, but the result will not be \(0\). If `bits` is set to \(0\), an exception will result.

void fmpz_randm(fmpz_t f, flint_rand_t state, const fmpz_t m)

  Generates a random integer in the range \(0\) to \(m - 1\) inclusive.

void fmpz_randtest_mod(fmpz_t f, flint_rand_t state, const fmpz_t m)

  Generates a random integer in the range \(0\) to \(m - 1\) inclusive, with an increased probability of generating values close to the endpoints.

void fmpz_randtest_mod_signed(fmpz_t f, flint_rand_t state, const fmpz_t m)

  Generates a random integer in the range \((-m/2, m/2]\), with an increased probability of generating values close to the endpoints or close to zero.

void fmpz_randprime(fmpz_t f, flint_rand_t state, flint_bitcnt_t bits, int proved)

  Generates a random prime number with the given number of bits.

  The generation is performed by choosing a random number and then finding the next largest prime, and therefore does not quite give a uniform distribution over the set of primes with that many bits.

  Random number generation is performed using the standard FLINT random number generator, which is not suitable for cryptographic use.

  If `proved` is nonzero, then the integer returned is guaranteed to actually be prime.

## Conversion

slong fmpz_get_si(const fmpz_t f)

  Returns \(f\) as a `slong`. The result is undefined if \(f\) does not fit into a `slong`.

ulong fmpz_get_ui(const fmpz_t f)

  Returns \(f\) as an `ulong`. The result is undefined if \(f\) does not fit into an `ulong` or is negative.

void fmpz_get_uiui(ulong *hi, ulong *low, const fmpz_t f)

  If \(f\) consists of two limbs, then `*hi` and `*low` are set to the high and low limbs, otherwise `*low` is set to the low limb and `*hi` is set to \(0\).

ulong fmpz_get_nmod(const fmpz_t f, nmod_t mod)

  Returns \(f\) modulo the integer given by \(mod\).

double fmpz_get_d(const fmpz_t f)

  Returns \(f\) as a `double`, rounding down towards zero if \(f\) cannot be represented exactly. The outcome is undefined if \(f\) is too large to fit in the normal range of a double.

void fmpz_set_mpf(fmpz_t f, const mpf_t x)

  Sets \(f\) to the `mpf_t` \(x\), rounding down towards zero if the value of \(x\) is fractional.

  Note: Requires that `gmp.h` has been included before any FLINT header is included.

void fmpz_get_mpf(mpf_t x, const fmpz_t f)

  Sets the value of the `mpf_t` \(x\) to the value of \(f\).

  Note: Requires that `gmp.h` has been included before any FLINT header is included.

void fmpz_get_mpfr(mpfr_t x, const fmpz_t f, mpfr_rnd_t rnd)

  Sets the value of \(x\) from \(f\), rounded toward the given direction `rnd`.

  Note: Requires that `mpfr.h` has been included before any FLINT header is included.

double fmpz_get_d_2exp(slong *exp, const fmpz_t f)

  Returns \(f\) as a normalized `double` along with a \(2\)-exponent `exp`, i.e. if \(r\) is the return value then \(f = r 2^{exp}\), to within 1 ULP.

void fmpz_get_mpz(mpz_t x, const fmpz_t f)

  Sets the `mpz_t` \(x\) to the same value as \(f\).

  Note: Requires that `gmp.h` has been included before any FLINT header is included.

int fmpz_get_mpn(nn_ptr *n, fmpz_t n_in)

  Sets the `nn_ptr` \(n\) to the same value as \(n_{in}\). Returned integer is number of limbs allocated to \(n\), minimum number of limbs required to hold the value stored in \(n_{in}\).

char *fmpz_get_str(char *str, int b, const fmpz_t f)

  Returns the representation of \(f\) in base \(b\), which can vary between \(2\) and \(62\), inclusive.

  If `str` is `NULL`, the result string is allocated by the function. Otherwise, it is up to the caller to ensure that the allocated block of memory is sufficiently large.

void fmpz_set_si(fmpz_t f, slong val)

  Sets \(f\) to the given `slong` value.

void fmpz_set_ui(fmpz_t f, ulong val)

  Sets \(f\) to the given `ulong` value.

void fmpz_set_d(fmpz_t f, double c)

  Sets \(f\) to the `double` \(c\), rounding down towards zero if the value of \(c\) is fractional. The outcome is undefined if \(c\) is infinite, not-a-number, or subnormal.

void fmpz_set_d_2exp(fmpz_t f, double d, slong exp)

  Sets \(f\) to the nearest integer to \(d 2^{exp}\).

void fmpz_neg_ui(fmpz_t f, ulong val)

  Sets \(f\) to the given `ulong` value, and then negates \(f\).

void fmpz_set_uiui(fmpz_t f, ulong hi, ulong lo)

  Sets \(f\) to `lo`, plus `hi` shifted to the left by `FLINT_BITS`.

void fmpz_neg_uiui(fmpz_t f, ulong hi, ulong lo)

  Sets \(f\) to `lo`, plus `hi` shifted to the left by `FLINT_BITS`, and then negates \(f\).

void fmpz_set_signed_uiui(fmpz_t f, ulong hi, ulong lo)

  Sets \(f\) to `lo`, plus `hi` shifted to the left by `FLINT_BITS`, interpreted as a signed two’s complement integer with `2 * FLINT_BITS` bits.

void fmpz_set_signed_uiuiui(fmpz_t f, ulong hi, ulong mid, ulong lo)

  Sets \(f\) to `lo`, plus `mid` shifted to the left by `FLINT_BITS`, plus `hi` shifted to the left by `2*FLINT_BITS` bits, interpreted as a signed two’s complement integer with `3 * FLINT_BITS` bits.

void fmpz_set_ui_array(fmpz_t out, const ulong *in, slong n)

  Sets `out` to the nonnegative integer `in[0] + in[1]*X + ... + in[n - 1]*X^(n - 1)` where `X = 2^FLINT_BITS`. It is assumed that `n > 0`.

void fmpz_set_signed_ui_array(fmpz_t out, const ulong *in, slong n)

  Sets `out` to the integer represented in `in[0], ..., in[n - 1]` as a signed two’s complement integer with `n * FLINT_BITS` bits. It is assumed that `n > 0`. The function operates as a call to `fmpz_set_ui_array()` followed by a symmetric remainder modulo \(2^{n\cdot FLINT\_BITS}\).

void fmpz_get_ui_array(ulong *out, slong n, const fmpz_t in)

  Assuming that the nonnegative integer `in` can be represented in the form `out[0] + out[1]*X + ... + out[n - 1]*X^(n - 1)`, where \(X = 2^{FLINT\_BITS}\), sets the corresponding elements of `out` so that this is true. It is assumed that `n > 0`.

void fmpz_get_signed_ui_array(ulong *out, slong n, const fmpz_t in)

  Retrieves the value of \(in\) modulo \(2^{n * FLINT\_BITS}\) and puts the \(n\) words of the result in `out[0], ..., out[n-1]`. This will give a signed two’s complement representation of \(in\) (assuming \(in\) doesn’t overflow the array).

void fmpz_set_mpn_large(fmpz_t z, nn_srcptr src, slong n, int negative)

  Sets z to the integer represented by the n limbs in the array src, or minus this value if negative is 1. Requires \(n \ge 2\) and that the top limb of src is nonzero. Note that fmpz_set_ui, fmpz_neg_ui can be used for single-limb integers.

void fmpz_get_signed_uiui(ulong *hi, ulong *lo, const fmpz_t in)

  Retrieves the value of \(in\) modulo \(2^{2 * FLINT\_BITS}\) and puts the high and low words into `*hi` and `*lo` respectively.

void fmpz_set_mpz(fmpz_t f, const mpz_t x)

  Sets \(f\) to the given `mpz_t` value.

  Note: Requires that `gmp.h` has been included before any FLINT header is included.

int fmpz_set_str(fmpz_t f, const char *str, int b)

  Sets \(f\) to the value given in the null-terminated string `str`, in base \(b\). The base \(b\) can vary between \(2\) and \(62\), inclusive. Returns \(0\) if the string contains a valid input and \(-1\) otherwise.

void fmpz_set_ui_smod(fmpz_t f, ulong x, ulong m)

  Sets \(f\) to the signed remainder \(y \equiv x \bmod m\) satisfying \(-m/2 < y \leq m/2\), given \(x\) which is assumed to satisfy \(0 \leq x < m\).

void flint_mpz_init_set_readonly(mpz_t z, const fmpz_t f)

  Sets the uninitialised `mpz_t` \(z\) to the value of the readonly `fmpz_t` \(f\).

  Note that it is assumed that \(f\) does not change during the lifetime of \(z\).

  The integer \(z\) has to be cleared by a call to `flint_mpz_clear_readonly()`.

  The suggested use of the two functions is as follows:

      fmpz_t f;
      ...
      {
          mpz_t z;

          flint_mpz_init_set_readonly(z, f);
          foo(..., z);
          flint_mpz_clear_readonly(z);
      }

  This provides a convenient function for user code, only requiring to work with the types `fmpz_t` and `mpz_t`.

  In critical code, the following approach may be favourable:

      fmpz_t f;
      ...
      {
          mpz_ptr z;

          z = _fmpz_promote_val(f);
          foo(..., z);
          _fmpz_demote_val(f);
      }

  Note: Requires that `gmp.h` has been included before any FLINT header is included.

void flint_mpz_clear_readonly(mpz_t z)

  Clears the readonly `mpz_t` \(z\).

  Note: Requires that `gmp.h` has been included before any FLINT header is included.

void fmpz_init_set_readonly(fmpz_t f, const mpz_t z)

  Sets the uninitialised `fmpz_t` \(f\) to a readonly version of the integer \(z\).

  Note that the value of \(z\) is assumed to remain constant throughout the lifetime of \(f\).

  The `fmpz_t` \(f\) has to be cleared by calling the function `fmpz_clear_readonly()`.

  The suggested use of the two functions is as follows:

      mpz_t z;
      ...
      {
          fmpz_t f;

          fmpz_init_set_readonly(f, z);
          foo(..., f);
          fmpz_clear_readonly(f);
      }

  Note: Requires that `gmp.h` has been included before any FLINT header is included.

void fmpz_clear_readonly(fmpz_t f)

  Clears the readonly `fmpz_t` \(f\).

## Input and output

int fmpz_read(fmpz_t f)

  Reads a multiprecision integer from `stdin`. The format is an optional minus sign, followed by one or more digits. The first digit should be non-zero unless it is the only digit.

  In case of success, returns a positive number. In case of failure, returns a non-positive number.

  This convention is adopted in light of the return values of `scanf` from the standard library and `mpz_inp_str` from GMP.

int fmpz_fread(FILE *file, fmpz_t f)

  Reads a multiprecision integer from the stream `file`. The format is an optional minus sign, followed by one or more digits. The first digit should be non-zero unless it is the only digit.

  In case of success, returns a positive number. In case of failure, returns a non-positive number.

  This convention is adopted in light of the return values of `scanf` from the standard library and `mpz_inp_str` from GMP.

size_t fmpz_inp_raw(fmpz_t x, FILE *fin)

  Reads a multiprecision integer from the stream `file`. The format is raw binary format write by `fmpz_out_raw()`.

  In case of success, return a positive number, indicating number of bytes read. In case of failure 0.

  This function calls the `mpz_inp_raw` function in library gmp. So that it can read the raw data written by `mpz_inp_raw` directly.

int fmpz_fprint(FILE *fs, const fmpz_t x)
int fmpz_print(const fmpz_t x)

  Prints the value \(x\) to `fs` or `stdout`, without a carriage return. The value is printed as either \(0\), the decimal digits of a positive integer, or a minus sign followed by the digits of a negative integer.

  Returns the number of characters written to file stream.

size_t fmpz_out_raw(FILE *fout, const fmpz_t x)

  Writes the value \(x\) to `file`. The value is written in raw binary format. The integer is written in 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 `fmpz_inp_raw`.

  In case of success, return a positive number, indicating number of bytes written. In case of failure, return 0.

  The output of this can also be read by `mpz_inp_raw` from GMP, since this function calls the `mpz_inp_raw` function in library gmp.

## Basic properties and manipulation

size_t fmpz_sizeinbase(const fmpz_t f, int b)

  Returns the size of the absolute value of \(f\) in base \(b\), measured in numbers of digits. The base \(b\) can be between \(2\) and \(62\), inclusive.

flint_bitcnt_t fmpz_bits(const fmpz_t f)

  Returns the number of bits required to store the absolute value of \(f\). If \(f\) is \(0\) then \(0\) is returned.

slong fmpz_size(const fmpz_t f)

  Returns the number of limbs required to store the absolute value of \(f\). If \(f\) is zero then \(0\) is returned.

int fmpz_sgn(const fmpz_t f)

  Returns \(-1\) if the sign of \(f\) is negative, \(+1\) if it is positive, otherwise returns \(0\).

flint_bitcnt_t fmpz_val2(const fmpz_t f)

  Returns the exponent of the largest power of two dividing \(f\), or equivalently the number of trailing zeros in the binary expansion of \(f\). If \(f\) is zero then \(0\) is returned.

void fmpz_swap(fmpz_t f, fmpz_t g)

  Efficiently swaps \(f\) and \(g\). No data is copied.

void fmpz_set(fmpz_t f, const fmpz_t g)

  Sets \(f\) to the same value as \(g\).

void fmpz_zero(fmpz_t f)

  Sets \(f\) to zero.

void fmpz_one(fmpz_t f)

  Sets \(f\) to one.

int fmpz_abs_fits_ui(const fmpz_t f)

  Returns whether the absolute value of \(f\) fits into an `ulong`.

int fmpz_fits_si(const fmpz_t f)

  Returns whether the value of \(f\) fits into a `slong`.

void fmpz_setbit(fmpz_t f, ulong i)

  Sets bit index \(i\) of \(f\).

int fmpz_tstbit(const fmpz_t f, ulong i)

  Test bit index \(i\) of \(f\) and return \(0\) or \(1\), accordingly.

ulong fmpz_abs_lbound_ui_2exp(slong *exp, const fmpz_t x, int bits)

  For nonzero \(x\), returns a mantissa \(m\) with exactly `bits` bits and sets `exp` to an exponent \(e\), such that \(|x| \ge m 2^e\). The number of bits must be between 1 and `FLINT_BITS` inclusive. The mantissa is guaranteed to be correctly rounded.

ulong fmpz_abs_ubound_ui_2exp(slong *exp, const fmpz_t x, int bits)

  For nonzero \(x\), returns a mantissa \(m\) with exactly `bits` bits and sets `exp` to an exponent \(e\), such that \(|x| \le m 2^e\). The number of bits must be between 1 and `FLINT_BITS` inclusive. The mantissa is either correctly rounded or one unit too large (possibly meaning that the exponent is one too large, if the mantissa is a power of two).

## Comparison

int fmpz_cmp(const fmpz_t f, const fmpz_t g)

int fmpz_cmp_ui(const fmpz_t f, ulong g)

int fmpz_cmp_si(const fmpz_t f, slong g)

  Returns a negative value if \(f < g\), positive value if \(g < f\), otherwise returns \(0\).

int fmpz_cmpabs(const fmpz_t f, const fmpz_t g)

  Returns a negative value if \(\lvert f\rvert < \lvert g\rvert\), positive value if \(\lvert g\rvert < \lvert f \rvert\), otherwise returns \(0\).

int fmpz_cmp2abs(const fmpz_t f, const fmpz_t g)

  Returns a negative value if \(\lvert f\rvert < \lvert 2g\rvert\), positive value if \(\lvert 2g\rvert < \lvert f \rvert\), otherwise returns \(0\).

int fmpz_equal(const fmpz_t f, const fmpz_t g)

int fmpz_equal_ui(const fmpz_t f, ulong g)

int fmpz_equal_si(const fmpz_t f, slong g)

  Returns \(1\) if \(f\) is equal to \(g\), otherwise returns \(0\).

int fmpz_is_zero(const fmpz_t f)

  Returns \(1\) if \(f\) is \(0\), otherwise returns \(0\).

int fmpz_is_one(const fmpz_t f)

  Returns \(1\) if \(f\) is equal to one, otherwise returns \(0\).

int fmpz_is_pm1(const fmpz_t f)

  Returns \(1\) if \(f\) is equal to one or minus one, otherwise returns \(0\).

int fmpz_is_even(const fmpz_t f)

  Returns whether the integer \(f\) is even.

int fmpz_is_odd(const fmpz_t f)

  Returns whether the integer \(f\) is odd.

## Basic arithmetic

void fmpz_neg(fmpz_t f1, const fmpz_t f2)

  Sets \(f_1\) to \(-f_2\).

void fmpz_abs(fmpz_t f1, const fmpz_t f2)

  Sets \(f_1\) to the absolute value of \(f_2\).

void fmpz_add(fmpz_t f, const fmpz_t g, const fmpz_t h)
void fmpz_add_ui(fmpz_t f, const fmpz_t g, ulong h)
void fmpz_add_si(fmpz_t f, const fmpz_t g, slong h)

  Sets \(f\) to \(g + h\).

void fmpz_sub(fmpz_t f, const fmpz_t g, const fmpz_t h)
void fmpz_sub_ui(fmpz_t f, const fmpz_t g, ulong h)
void fmpz_sub_si(fmpz_t f, const fmpz_t g, slong h)

  Sets \(f\) to \(g - h\).

void fmpz_mul(fmpz_t f, const fmpz_t g, const fmpz_t h)
void fmpz_mul_ui(fmpz_t f, const fmpz_t g, ulong h)
void fmpz_mul_si(fmpz_t f, const fmpz_t g, slong h)

  Sets \(f\) to \(g \times h\).

void fmpz_mul2_uiui(fmpz_t f, const fmpz_t g, ulong x, ulong y)

  Sets \(f\) to \(g \times x \times y\) where \(x\) and \(y\) are of type `ulong`.

void fmpz_mul_2exp(fmpz_t f, const fmpz_t g, ulong e)

  Sets \(f\) to \(g \times 2^e\).

  Note: Assumes that `e + FLINT_BITS` does not overflow.

void fmpz_one_2exp(fmpz_t f, ulong e)

  Sets \(f\) to \(2^e\).

void fmpz_addmul(fmpz_t f, const fmpz_t g, const fmpz_t h)
void fmpz_addmul_ui(fmpz_t f, const fmpz_t g, ulong h)
void fmpz_addmul_si(fmpz_t f, const fmpz_t g, slong h)

  Sets \(f\) to \(f + g \times h\).

void fmpz_submul(fmpz_t f, const fmpz_t g, const fmpz_t h)
void fmpz_submul_ui(fmpz_t f, const fmpz_t g, ulong h)
void fmpz_submul_si(fmpz_t f, const fmpz_t g, slong h)

  Sets \(f\) to \(f - g \times h\).

void fmpz_fmma(fmpz_t f, const fmpz_t a, const fmpz_t b, const fmpz_t c, const fmpz_t d)

  Sets \(f\) to \(a \times b + c \times d\).

void fmpz_fmms(fmpz_t f, const fmpz_t a, const fmpz_t b, const fmpz_t c, const fmpz_t d)

  Sets \(f\) to \(a \times b - c \times d\).

void fmpz_cdiv_qr(fmpz_t f, fmpz_t s, const fmpz_t g, const fmpz_t h)

void fmpz_fdiv_qr(fmpz_t f, fmpz_t s, const fmpz_t g, const fmpz_t h)

void fmpz_tdiv_qr(fmpz_t f, fmpz_t s, const fmpz_t g, const fmpz_t h)

void fmpz_ndiv_qr(fmpz_t f, fmpz_t s, const fmpz_t g, const fmpz_t h)

void fmpz_cdiv_q(fmpz_t f, const fmpz_t g, const fmpz_t h)

void fmpz_fdiv_q(fmpz_t f, const fmpz_t g, const fmpz_t h)

void fmpz_tdiv_q(fmpz_t f, const fmpz_t g, const fmpz_t h)

void fmpz_cdiv_q_si(fmpz_t f, const fmpz_t g, slong h)

void fmpz_fdiv_q_si(fmpz_t f, const fmpz_t g, slong h)

void fmpz_tdiv_q_si(fmpz_t f, const fmpz_t g, slong h)

void fmpz_cdiv_q_ui(fmpz_t f, const fmpz_t g, ulong h)

void fmpz_fdiv_q_ui(fmpz_t f, const fmpz_t g, ulong h)

void fmpz_tdiv_q_ui(fmpz_t f, const fmpz_t g, ulong h)

void fmpz_cdiv_q_2exp(fmpz_t f, const fmpz_t g, ulong exp)

void fmpz_fdiv_q_2exp(fmpz_t f, const fmpz_t g, ulong exp)

void fmpz_tdiv_q_2exp(fmpz_t f, const fmpz_t g, ulong exp)

void fmpz_fdiv_r(fmpz_t s, const fmpz_t g, const fmpz_t h)

void fmpz_cdiv_r_2exp(fmpz_t s, const fmpz_t g, ulong exp)

void fmpz_fdiv_r_2exp(fmpz_t s, const fmpz_t g, ulong exp)

void fmpz_tdiv_r_2exp(fmpz_t s, const fmpz_t g, ulong exp)

  Sets \(f\) to the quotient of \(g\) by \(h\) and/or \(s\) to the remainder. For the `2exp` functions, `h = 2^exp`. If \(h\) is \(0\) an exception is raised.

  Rounding is made in the following way:

  - `fdiv` rounds the quotient via floor rounding.

  - `cdiv` rounds the quotient via ceil rounding.

  - `tdiv` rounds the quotient via truncation, i.e. rounding towards zero.

  - `ndiv` rounds the quotient such that the remainder has the smallest absolute value. In case of ties, it rounds the quotient towards zero.

ulong fmpz_cdiv_ui(const fmpz_t g, ulong h)

ulong fmpz_fdiv_ui(const fmpz_t g, ulong h)

ulong fmpz_tdiv_ui(const fmpz_t g, ulong h)

  Returns the absolute value remainder of \(g\) divided by \(h\), following the convention of rounding as seen above. If \(h\) is zero an exception is raised.

void fmpz_divexact(fmpz_t f, const fmpz_t g, const fmpz_t h)

void fmpz_divexact_si(fmpz_t f, const fmpz_t g, slong h)

void fmpz_divexact_ui(fmpz_t f, const fmpz_t g, ulong h)

  Sets \(f\) to the quotient of \(g\) and \(h\), assuming that the division is exact, i.e. \(g\) is a multiple of \(h\). If \(h\) is \(0\) an exception is raised.

void fmpz_divexact2_uiui(fmpz_t f, const fmpz_t g, ulong x, ulong y)

  Sets \(f\) to the quotient of \(g\) and \(h = x \times y\), assuming that the division is exact, i.e. \(g\) is a multiple of \(h\). If \(x\) or \(y\) is \(0\) an exception is raised.

int fmpz_divisible(const fmpz_t f, const fmpz_t g)

int fmpz_divisible_si(const fmpz_t f, slong g)

  Returns \(1\) if there is an integer \(q\) with \(f = q g\) and \(0\) if there is none.

int fmpz_divides(fmpz_t q, const fmpz_t f, const fmpz_t g)

  Returns \(1\) if there is an integer \(q\) with \(f = q g\) and sets \(q\) to the quotient. Otherwise returns \(0\) and sets \(q\) to \(0\).

void fmpz_mod(fmpz_t f, const fmpz_t g, const fmpz_t h)

  Sets \(f\) to the remainder of \(g\) divided by \(h\) such that the remainder is non-negative. Assumes that \(h\) is not zero.

ulong fmpz_mod_ui(fmpz_t f, const fmpz_t g, ulong h)

  Sets \(f\) to the remainder of \(g\) divided by \(h\) such that the remainder is non-negative and also returns this value. Raises an exception if \(h\) is zero.

void fmpz_smod(fmpz_t f, const fmpz_t g, const fmpz_t h)

  Sets \(f\) to the signed remainder \(y \equiv g \bmod h\) satisfying \(-\lvert h \rvert/2 < y \leq \lvert h\rvert/2\).

void fmpz_preinvn_init(fmpz_preinvn_t inv, const fmpz_t f)

  Compute a precomputed inverse `inv` of `f` for use in the `preinvn` functions listed below.

void fmpz_preinvn_clear(fmpz_preinvn_t inv)

  Clean up the resources used by a precomputed inverse created with the `fmpz_preinvn_init()` function.

void fmpz_fdiv_qr_preinvn(fmpz_t f, fmpz_t s, const fmpz_t g, const fmpz_t h, const fmpz_preinvn_t hinv)

  As per `fmpz_fdiv_qr()`, but takes a precomputed inverse `hinv` of \(h\) constructed using `fmpz_preinvn()`.

void fmpz_pow_ui(fmpz_t f, const fmpz_t g, ulong x)
void fmpz_ui_pow_ui(fmpz_t f, ulong g, ulong x)

  Sets \(f\) to \(g^x\). Defines \(0^0 = 1\).

int fmpz_pow_fmpz(fmpz_t f, const fmpz_t g, const fmpz_t x)

  Sets \(f\) to \(g^x\). Defines \(0^0 = 1\). Return \(1\) for success and \(0\) for failure. The function throws only if \(x\) is negative.

void fmpz_powm_ui(fmpz_t f, const fmpz_t g, ulong e, const fmpz_t m)

void fmpz_powm(fmpz_t f, const fmpz_t g, const fmpz_t e, const fmpz_t m)

  Sets \(f\) to \(g^e \bmod{m}\). If \(e = 0\), sets \(f\) to \(1\).

  Assumes that \(m \neq 0\), raises an `abort` signal otherwise.

slong fmpz_clog(const fmpz_t x, const fmpz_t b)
slong fmpz_clog_ui(const fmpz_t x, ulong b)

  Returns \(\lceil\log_b x\rceil\).

  Assumes that \(x \geq 1\) and \(b \geq 2\) and that the return value fits into a signed `slong`.

slong fmpz_flog(const fmpz_t x, const fmpz_t b)
slong fmpz_flog_ui(const fmpz_t x, ulong b)

  Returns \(\lfloor\log_b x\rfloor\).

  Assumes that \(x \geq 1\) and \(b \geq 2\) and that the return value fits into a signed `slong`.

double fmpz_dlog(const fmpz_t x)

  Returns a double precision approximation of the natural logarithm of \(x\).

  The accuracy depends on the implementation of the floating-point logarithm provided by the C standard library. The result can typically be expected to have a relative error no greater than 1-2 bits.

int fmpz_sqrtmod(fmpz_t b, const fmpz_t a, const fmpz_t p)

  If \(p\) is prime, set \(b\) to a square root of \(a\) modulo \(p\) if \(a\) is a quadratic residue modulo \(p\) and return \(1\), otherwise return \(0\).

  If \(p\) is not prime the return value is with high probability \(0\), indicating that \(p\) is not prime, or \(a\) is not a square modulo \(p\). If \(p\) is not prime and the return value is \(1\), the value of \(b\) is meaningless.

void fmpz_sqrt(fmpz_t f, const fmpz_t g)

  Sets \(f\) to the integer part of the square root of \(g\), where \(g\) is assumed to be non-negative. If \(g\) is negative, an exception is raised.

void fmpz_sqrtrem(fmpz_t f, fmpz_t r, const fmpz_t g)

  Sets \(f\) to the integer part of the square root of \(g\), where \(g\) is assumed to be non-negative, and sets \(r\) to the remainder, that is, the difference \(g - f^2\). If \(g\) is negative, an exception is raised. The behaviour is undefined if \(f\) and \(r\) are aliases.

int fmpz_is_square(const fmpz_t f)

  Returns nonzero if \(f\) is a perfect square and zero otherwise.

int fmpz_root(fmpz_t r, const fmpz_t f, slong n)

  Set \(r\) to the integer part of the \(n\)-th root of \(f\). Requires that \(n > 0\) and that if \(n\) is even then \(f\) be non-negative, otherwise an exception is raised. The function returns \(1\) if the root was exact, otherwise \(0\).

int fmpz_is_perfect_power(fmpz_t root, const fmpz_t f)

  If \(f\) is a perfect power \(r^k\) set `root` to \(r\) and return \(k\), otherwise return \(0\). Note that \(-1, 0, 1\) are all considered perfect powers. No guarantee is made about \(r\) or \(k\) being the smallest possible value. Negative values of \(f\) are permitted.

void fmpz_fac_ui(fmpz_t f, ulong n)

  Sets \(f\) to the factorial \(n!\) where \(n\) is an `ulong`.

void fmpz_fib_ui(fmpz_t f, ulong n)

  Sets \(f\) to the Fibonacci number \(F_n\) where \(n\) is an `ulong`.

void fmpz_bin_uiui(fmpz_t f, ulong n, ulong k)

  Sets \(f\) to the binomial coefficient \({n \choose k}\).

void _fmpz_rfac_ui(fmpz_t r, const fmpz_t x, ulong a, ulong b)

  Sets \(r\) to the rising factorial \((x+a) (x+a+1) (x+a+2) \cdots (x+b-1)\). Assumes \(b > a\).

void fmpz_rfac_ui(fmpz_t r, const fmpz_t x, ulong k)

  Sets \(r\) to the rising factorial \(x (x+1) (x+2) \cdots (x+k-1)\).

void fmpz_rfac_uiui(fmpz_t r, ulong x, ulong k)

  Sets \(r\) to the rising factorial \(x (x+1) (x+2) \cdots (x+k-1)\).

void fmpz_mul_tdiv_q_2exp(fmpz_t f, const fmpz_t g, const fmpz_t h, ulong exp)

  Sets \(f\) to the product of \(g\) and \(h\) divided by `2^exp`, rounding down towards zero.

void fmpz_mul_si_tdiv_q_2exp(fmpz_t f, const fmpz_t g, slong x, ulong exp)

  Sets \(f\) to the product of \(g\) and \(x\) divided by `2^exp`, rounding down towards zero.

## Greatest common divisor

void fmpz_gcd_ui(fmpz_t f, const fmpz_t g, ulong h)

void fmpz_gcd(fmpz_t f, const fmpz_t g, const fmpz_t h)

  Sets \(f\) to the greatest common divisor of \(g\) and \(h\). The result is always non-negative, even if one of \(g\) and \(h\) is negative.

void fmpz_gcd3(fmpz_t f, const fmpz_t a, const fmpz_t b, const fmpz_t c)

  Sets \(f\) to the greatest common divisor of \(a\), \(b\) and \(c\). This is equivalent to calling `fmpz_gcd` twice, but may be faster.

void fmpz_lcm(fmpz_t f, const fmpz_t g, const fmpz_t h)

  Sets \(f\) to the least common multiple of \(g\) and \(h\). The result is always nonnegative, even if one of \(g\) and \(h\) is negative.

void fmpz_gcdinv(fmpz_t d, fmpz_t a, const fmpz_t f, const fmpz_t g)

  Given integers \(f, g\) with \(0 \leq f < g\), computes the greatest common divisor \(d = \gcd(f, g)\) and the modular inverse \(a = f^{-1} \pmod{g}\), whenever \(f \neq 0\).

  Assumes that \(d\) and \(a\) are not aliased.

void fmpz_xgcd(fmpz_t d, fmpz_t a, fmpz_t b, const fmpz_t f, const fmpz_t g)

  Computes the extended GCD of \(f\) and \(g\), i.e. the values \(a\) and \(b\) such that \(af + bg = d\), where \(d = \gcd(f, g)\). Here \(a\) will be the same as calling `fmpz_gcdinv` when \(f < g\) (or vice versa for \(b\) when \(g < f\)).

  To obtain the canonical solution to Bézout’s identity, call `fmpz_xgcd_canonical_bezout` instead. This is also faster.

  Assumes that there is no aliasing among the outputs.

void fmpz_xgcd_canonical_bezout(fmpz_t d, fmpz_t a, fmpz_t b, const fmpz_t f, const fmpz_t g)

  Computes the extended GCD \(\operatorname{xgcd}(f, g) = (d, a, b)\) such that the solution is the canonical solution to Bézout’s identity. We define the canonical solution to satisfy one of the following if one of the given conditions apply:

  \[ \begin{align}\begin{aligned}\operatorname{xgcd}(\pm g, g) &= \bigl(|g|, 0, \operatorname{sgn}(g)\bigr)\\\operatorname{xgcd}(f, 0) &= \bigl(|f|, \operatorname{sgn}(f), 0\bigr)\\\operatorname{xgcd}(0, g) &= \bigl(|g|, 0, \operatorname{sgn}(g)\bigr)\\\operatorname{xgcd}(f, \mp 1) &= (1, 0, \mp 1)\\\operatorname{xgcd}(\mp 1, g) &= (1, \mp 1, 0)\quad g \neq 0, \pm 1\\\operatorname{xgcd}(\mp 2 d, g) &= \bigl(d, {\textstyle\frac{d - |g|}{\mp 2 d}}, \operatorname{sgn}(g)\bigr)\\\operatorname{xgcd}(f, \mp 2 d) &= \bigl(d, \operatorname{sgn}(f), {\textstyle\frac{d - |g|}{\mp 2 d}}\bigr).\end{aligned}\end{align} \]

  If the pair \((f, g)\) does not satisfy any of these conditions, the solution \((d, a, b)\) will satisfy the following:

  \[|a| < \Bigl| \frac{g}{2 d} \Bigr|, \qquad |b| < \Bigl| \frac{f}{2 d} \Bigr|.\]

  Assumes that there is no aliasing among the outputs.

void fmpz_xgcd_partial(fmpz_t co2, fmpz_t co1, fmpz_t r2, fmpz_t r1, const fmpz_t L)

  This function is an implementation of Lehmer extended GCD with early termination, as used in the `qfb` module. It terminates early when remainders fall below the specified bound. The initial values `r1` and `r2` are treated as successive remainders in the Euclidean algorithm and are replaced with the last two remainders computed. The values `co1` and `co2` are the last two cofactors and satisfy the identity `co2*r1 - co1*r2 == +/- r2_orig` upon termination, where `r2_orig` is the starting value of `r2` supplied, and `r1` and `r2` are the final values.

  Aliasing of inputs is not allowed. Similarly aliasing of inputs and outputs is not allowed.

## Modular arithmetic

slong _fmpz_remove(fmpz_t x, const fmpz_t f, double finv)

  Removes all factors \(f\) from \(x\) and returns the number of such.

  Assumes that \(x\) is non-zero, that \(f > 1\) and that `finv` is the precomputed `double` inverse of \(f\) whenever \(f\) is a small integer and \(0\) otherwise.

  Does not support aliasing.

slong fmpz_remove(fmpz_t rop, const fmpz_t op, const fmpz_t f)

  Remove all occurrences of the factor \(f > 1\) from the integer `op` and sets `rop` to the resulting integer.

  If `op` is zero, sets `rop` to `op` and returns \(0\).

  Returns an `abort` signal if any of the assumptions are violated.

int fmpz_invmod(fmpz_t f, const fmpz_t g, const fmpz_t h)

  Sets \(f\) to the inverse of \(g\) modulo \(h\). The value of \(h\) may not be \(0\) otherwise an exception results. If the inverse exists the return value will be non-zero, otherwise the return value will be \(0\) and the value of \(f\) undefined. As a special case, we consider any number invertible modulo \(h = \pm 1\), with inverse 0.

void fmpz_negmod(fmpz_t f, const fmpz_t g, const fmpz_t h)

  Sets \(f\) to \(-g \pmod{h}\), assuming \(g\) is reduced modulo \(h\).

int fmpz_jacobi(const fmpz_t a, const fmpz_t n)

  Computes the Jacobi symbol \(\left(\frac{a}{n}\right)\) for any \(a\) and odd positive \(n\).

int fmpz_kronecker(const fmpz_t a, const fmpz_t n)

  Computes the Kronecker symbol \(\left(\frac{a}{n}\right)\) for any \(a\) and any \(n\).

void fmpz_divides_mod_list(fmpz_t xstart, fmpz_t xstride, fmpz_t xlength, const fmpz_t a, const fmpz_t b, const fmpz_t n)

  Set \(xstart\), \(xstride\), and \(xlength\) so that the solution set for \(x\) modulo \(n\) in \(a x = b \bmod n\) is exactly \(\{xstart + xstride\,i \mid 0 \le i < xlength\}\). This function essentially gives a list of possibilities for the fraction \(a/b\) modulo \(n\). The outputs may not be aliased, and \(n\) should be positive.

## Bit packing and unpacking

int fmpz_bit_pack(ulong *arr, flint_bitcnt_t shift, flint_bitcnt_t bits, const fmpz_t coeff, int negate, int borrow)

  Shifts the given coefficient to the left by `shift` bits and adds it to the integer in `arr` in a field of the given number of bits:

      shift  bits  --------------

      X X X C C C C 0 0 0 0 0 0 0

  An optional borrow of \(1\) can be subtracted from `coeff` before it is packed. If `coeff` is negative after the borrow, then a borrow will be returned by the function.

  The value of `shift` is assumed to be less than `FLINT_BITS`. All but the first `shift` bits of `arr` are assumed to be zero on entry to the function.

  The value of `coeff` may also be optionally (and notionally) negated before it is used, by setting the `negate` parameter to \(-1\).

int fmpz_bit_unpack(fmpz_t coeff, ulong *arr, flint_bitcnt_t shift, flint_bitcnt_t bits, int negate, int borrow)

  A bit field of the given number of bits is extracted from `arr`, starting after `shift` bits, and placed into `coeff`. An optional borrow of \(1\) may be added to the coefficient. If the result is negative, a borrow of \(1\) is returned. Finally, the resulting `coeff` may be negated by setting the `negate` parameter to \(-1\).

  The value of `shift` is expected to be less than `FLINT_BITS`.

void fmpz_bit_unpack_unsigned(fmpz_t coeff, const ulong *arr, flint_bitcnt_t shift, flint_bitcnt_t bits)

  A bit field of the given number of bits is extracted from `arr`, starting after `shift` bits, and placed into `coeff`.

  The value of `shift` is expected to be less than `FLINT_BITS`.

## Logic Operations

void fmpz_complement(fmpz_t r, const fmpz_t f)

  The variable `r` is set to the ones-complement of `f`.

void fmpz_clrbit(fmpz_t f, ulong i)

  Sets the `i`th bit in `f` to zero.

void fmpz_combit(fmpz_t f, ulong i)

  Complements the `i`th bit in `f`.

void fmpz_and(fmpz_t r, const fmpz_t a, const fmpz_t b)

  Sets `r` to the bit-wise logical `and` of `a` and `b`.

void fmpz_or(fmpz_t r, const fmpz_t a, const fmpz_t b)

  Sets `r` to the bit-wise logical (inclusive) `or` of `a` and `b`.

void fmpz_xor(fmpz_t r, const fmpz_t a, const fmpz_t b)

  Sets `r` to the bit-wise logical exclusive `or` of `a` and `b`.

ulong fmpz_popcnt(const fmpz_t a)

  Returns the number of ‘1’ bits in the given Z (aka Hamming weight or population count). The return value is undefined if the input is negative.

## Chinese remaindering

The following functions can be used to reconstruct an integer from its residues modulo a set of prime numbers. The first two functions, `fmpz_CRT_ui()` and `fmpz_CRT()`, are easy to use and allow building the result one residue at a time, which is useful when the number of needed primes is not known in advance. The remaining functions support performing the modular reductions and reconstruction using balanced subdivision. This greatly improves efficiency for large integers but assumes that the basis of primes is known in advance. The user must precompute a `comb` structure and temporary working space with `fmpz_comb_init()` and `fmpz_comb_temp_init()`, and free this data afterwards. For simple demonstration programs showing how to use the CRT functions, see `crt.c` and `multi_crt.c` in the `examples` directory. The `fmpz_multi_CRT` class is similar to `fmpz_multi_CRT_ui` except that it performs error checking and works with arbitrary moduli.

void fmpz_CRT_ui(fmpz_t out, const fmpz_t r1, const fmpz_t m1, ulong r2, ulong m2, int sign)

  Uses the Chinese Remainder Theorem to compute the unique integer \(0 \le x < M\) (if sign = 0) or \(-M/2 < x \le M/2\) (if sign = 1) congruent to \(r_1\) modulo \(m_1\) and \(r_2\) modulo \(m_2\), where \(M = m_1 \times m_2\). The result \(x\) is stored in `out`.

  It is assumed that \(m_1\) and \(m_2\) are positive coprime integers.

  If sign = 0, it is assumed that \(0 \le r_1 < m_1\) and \(0 \le r_2 < m_2\). Otherwise, it is assumed that \(-m_1 \le r_1 < m_1\) and \(0 \le r_2 < m_2\).

void fmpz_CRT(fmpz_t out, const fmpz_t r1, const fmpz_t m1, const fmpz_t r2, const fmpz_t m2, int sign)

  Use the Chinese Remainder Theorem to set `out` to the unique value \(0 \le x < M\) (if sign = 0) or \(-M/2 < x \le M/2\) (if sign = 1) congruent to \(r_1\) modulo \(m_1\) and \(r_2\) modulo \(m_2\), where \(M = m_1 \times m_2\).

  It is assumed that \(m_1\) and \(m_2\) are positive coprime integers.

  If sign = 0, it is assumed that \(0 \le r_1 < m_1\) and \(0 \le r_2 < m_2\). Otherwise, it is assumed that \(-m_1 \le r_1 < m_1\) and \(0 \le r_2 < m_2\).

void fmpz_multi_mod_ui(ulong *out, const fmpz_t in, const fmpz_comb_t comb, fmpz_comb_temp_t temp)

  Reduces the multiprecision integer `in` modulo each of the primes stored in the `comb` structure. The array `out` will be filled with the residues modulo these primes. The structure `temp` is temporary space which must be provided by `fmpz_comb_temp_init()` and cleared by `fmpz_comb_temp_clear()`.

void fmpz_multi_CRT_ui(fmpz_t output, nn_srcptr residues, const fmpz_comb_t comb, fmpz_comb_temp_t ctemp, int sign)

  This function takes a set of residues modulo the list of primes contained in the `comb` structure and reconstructs a multiprecision integer modulo the product of the primes which has these residues modulo the corresponding primes.

  If \(N\) is the product of all the primes then `out` is normalised to be in the range \([0, N)\) if sign = 0 and the range \([-(N-1)/2, N/2]\) if sign = 1. The array `temp` is temporary space which must be provided by `fmpz_comb_temp_init()` and cleared by `fmpz_comb_temp_clear()`.

void fmpz_comb_init(fmpz_comb_t comb, nn_srcptr primes, slong num_primes)

  Initialises a `comb` structure for multimodular reduction and recombination. The array `primes` is assumed to contain `num_primes` primes each of `FLINT_BITS - 1` bits. Modular reductions and recombinations will be done modulo this list of primes. The `primes` array must not be `free`’d until the `comb` structure is no longer required and must be cleared by the user.

void fmpz_comb_temp_init(fmpz_comb_temp_t temp, const fmpz_comb_t comb)

  Creates temporary space to be used by multimodular and CRT functions based on an initialised `comb` structure.

void fmpz_comb_clear(fmpz_comb_t comb)

  Clears the given `comb` structure, releasing any memory it uses.

void fmpz_comb_temp_clear(fmpz_comb_temp_t temp)

  Clears temporary space `temp` used by multimodular and CRT functions using the given `comb` structure.

void fmpz_multi_CRT_init(fmpz_multi_CRT_t CRT)

  Initialize `CRT` for Chinese remaindering.

int fmpz_multi_CRT_precompute(fmpz_multi_CRT_t CRT, const fmpz *moduli, slong len)

  Configure `CRT` for repeated Chinese remaindering of `moduli`. The number of moduli, `len`, should be positive. A return of `0` indicates that the compilation failed and future calls to `fmpz_multi_CRT_precomp()` will leave the output undefined. A return of `1` indicates that the compilation was successful, which occurs if and only if either (1) `len == 1` and `modulus + 0` is nonzero, or (2) no modulus is \(0,1,-1\) and all moduli are pairwise relatively prime.

void fmpz_multi_CRT_precomp(fmpz_t output, const fmpz_multi_CRT_t P, const fmpz *inputs, int sign)

  Set `output` to an integer of smallest absolute value that is congruent to `values + i` modulo the `moduli + i` in `P`.

int fmpz_multi_CRT(fmpz_t output, const fmpz *moduli, const fmpz *values, slong len, int sign)

  Perform the same operation as `fmpz_multi_CRT_precomp()` while internally constructing and destroying the precomputed data. All of the remarks in `fmpz_multi_CRT_precompute()` apply.

void fmpz_multi_CRT_clear(fmpz_multi_CRT_t P)

  Free all space used by `CRT`.

## Primality testing

int fmpz_is_strong_probabprime(const fmpz_t n, const fmpz_t a)

  Returns \(1\) if \(n\) is a strong probable prime to base \(a\), otherwise it returns \(0\).

int fmpz_is_probabprime_lucas(const fmpz_t n)

  Performs a Lucas probable prime test with parameters chosen by Selfridge’s method \(A\) as per [BaiWag1980].

  Return \(1\) if \(n\) is a Lucas probable prime, otherwise return \(0\). This function declares some composites probably prime, but no primes composite.

int fmpz_is_probabprime_BPSW(const fmpz_t n)

  Perform a Baillie-PSW probable prime test with parameters chosen by Selfridge’s method \(A\) as per [BaiWag1980].

  Return \(1\) if \(n\) is a Lucas probable prime, otherwise return \(0\).

  There are no known composites passed as prime by this test, though infinitely many probably exist. The test will declare no primes composite.

int fmpz_is_probabprime(const fmpz_t p)

  Performs some trial division and then some probabilistic primality tests. If \(p\) is definitely composite, the function returns \(0\), otherwise it is declared probably prime, i.e. prime for most practical purposes, and the function returns \(1\). The chance of declaring a composite prime is very small.

  Subsequent calls to the same function do not increase the probability of the number being prime.

int fmpz_is_prime_pseudosquare(const fmpz_t n)

  Return \(0\) is \(n\) is composite. If \(n\) is too large (greater than about \(94\) bits) the function fails silently and returns \(-1\), otherwise, if \(n\) is proven prime by the pseudosquares method, return \(1\).

  Tests if \(n\) is a prime according to Theorem 2.7 in [LukPatWil1996].

  We first factor \(N\) using trial division up to some limit \(B\). In fact, the number of primes used in the trial factoring is at most `FLINT_PSEUDOSQUARES_CUTOFF`.

  Next we compute \(N/B\) and find the next pseudosquare \(L_p\) above this value, using a static table as per https://oeis.org/A002189/b002189.txt.

  As noted in the text, if \(p\) is prime then Step 3 will pass. This test rejects many composites, and so by this time we suspect that \(p\) is prime. If \(N\) is \(3\) or \(7\) modulo \(8\), we are done, and \(N\) is prime.

  We now run a probable prime test, for which no known counterexamples are known, to reject any composites. We then proceed to prove \(N\) prime by executing Step 4. In the case that \(N\) is \(1\) modulo \(8\), if Step 4 fails, we extend the number of primes \(p_i\) at Step 3 and hope to find one which passes Step 4. We take the test one past the largest \(p\) for which we have pseudosquares \(L_p\) tabulated, as this already corresponds to the next \(L_p\) which is bigger than \(2^{64}\) and hence larger than any prime we might be testing.

  As explained in the text, Condition 4 cannot fail if \(N\) is prime.

  The possibility exists that the probable prime test declares a composite prime. However in that case an error is printed, as that would be of independent interest.

int fmpz_is_prime_pocklington(fmpz_t F, fmpz_t R, const fmpz_t n, nn_ptr pm1, slong num_pm1)

  Applies the Pocklington primality test. The test computes a product \(F\) of prime powers which divide \(n - 1\).

  The function then returns either \(0\) if \(n\) is definitely composite or it returns \(1\) if all factors of \(n\) are \(1 \pmod{F}\). Also in that case, \(R\) is set to \((n - 1)/F\).

  NB: a return value of \(1\) only proves \(n\) prime if \(F \ge \sqrt{n}\).

  The function does not compute which primes divide \(n - 1\). Instead, these must be supplied as an array `pm1` of length `num_pm1`. It does not matter how many prime factors are supplied, but the more that are supplied, the larger F will be.

  There is a balance between the amount of time spent looking for factors of \(n - 1\) and the usefulness of the output (\(F\) may be as low as \(2\) in some cases).

  A reasonable heuristic seems to be to choose `limit` to be some small multiple of \(\log^3(n)/10\) (e.g. \(1, 2, 5\) or \(10\)) depending on how long one is prepared to wait, then to trial factor up to the limit. (See `_fmpz_nm1_trial_factors`.)

  Requires \(n\) to be odd.

void _fmpz_nm1_trial_factors(const fmpz_t n, nn_ptr pm1, slong *num_pm1, ulong limit)

  Trial factors \(n - 1\) up to the given limit (approximately) and stores the factors in an array `pm1` whose length is written out to `num_pm1`.

  One can use \(\log(n) + 2\) as a bound on the number of factors which might be produced (and hence on the length of the array that needs to be supplied).

int fmpz_is_prime_morrison(fmpz_t F, fmpz_t R, const fmpz_t n, nn_ptr pp1, slong num_pp1)

  Applies the Morrison \(p + 1\) primality test. The test computes a product \(F\) of primes which divide \(n + 1\).

  The function then returns either \(0\) if \(n\) is definitely composite or it returns \(1\) if all factors of \(n\) are \(\pm 1 \pmod{F}\). Also in that case, \(R\) is set to \((n + 1)/F\).

  NB: a return value of \(1\) only proves \(n\) prime if \(F > \sqrt{n} + 1\).

  The function does not compute which primes divide \(n + 1\). Instead, these must be supplied as an array `pp1` of length `num_pp1`. It does not matter how many prime factors are supplied, but the more that are supplied, the larger \(F\) will be.

  There is a balance between the amount of time spent looking for factors of \(n + 1\) and the usefulness of the output (\(F\) may be as low as \(2\) in some cases).

  A reasonable heuristic seems to be to choose `limit` to be some small multiple of \(\log^3(n)/10\) (e.g. \(1, 2, 5\) or \(10\)) depending on how long one is prepared to wait, then to trial factor up to the limit. (See `_fmpz_np1_trial_factors`.)

  Requires \(n\) to be odd and non-square.

void _fmpz_np1_trial_factors(const fmpz_t n, nn_ptr pp1, slong *num_pp1, ulong limit)

  Trial factors \(n + 1\) up to the given limit (approximately) and stores the factors in an array `pp1` whose length is written out to `num_pp1`.

  One can use \(\log(n) + 2\) as a bound on the number of factors which might be produced (and hence on the length of the array that needs to be supplied).

int fmpz_is_prime(const fmpz_t n)

  Attempts to prove \(n\) prime. If \(n\) is proven prime, the function returns \(1\). If \(n\) is definitely composite, the function returns \(0\).

  This function calls `n_is_prime()` for \(n\) that fits in a single word. For \(n\) larger than one word, it tests divisibility by a few small primes and whether \(n\) is a perfect square to rule out trivial composites. For \(n\) up to about 81 bits, it then uses a strong probable prime test (Miller-Rabin test) with the first 13 primes as witnesses. This has been shown to prove primality [SorWeb2016].

  For larger \(n\), it does a single base-2 strong probable prime test to eliminate most composite numbers. If \(n\) passes, it does a combination of Pocklington, Morrison and Brillhart, Lehmer, Selfridge tests. If any of these tests fails to give a proof, it falls back to performing an APRCL test.

  The APRCL test could theoretically fail to prove that \(n\) is prime or composite. In that case, the program aborts. This is not expected to occur in practice.

void fmpz_lucas_chain(fmpz_t Vm, fmpz_t Vm1, const fmpz_t A, const fmpz_t m, const fmpz_t n)

  Given \(V_0 = 2\), \(V_1 = A\) compute \(V_m, V_{m + 1} \pmod{n}\) from the recurrences \(V_j = AV_{j - 1} - V_{j - 2} \pmod{n}\).

  This is computed efficiently using \(V_{2j} = V_j^2 - 2 \pmod{n}\) and \(V_{2j + 1} = V_jV_{j + 1} - A \pmod{n}\).

  No aliasing is permitted.

void fmpz_lucas_chain_full(fmpz_t Vm, fmpz_t Vm1, const fmpz_t A, const fmpz_t B, const fmpz_t m, const fmpz_t n)

  Given \(V_0 = 2\), \(V_1 = A\) compute \(V_m, V_{m + 1} \pmod{n}\) from the recurrences \(V_j = AV_{j - 1} - BV_{j - 2} \pmod{n}\).

  This is computed efficiently using double and add formulas.

  No aliasing is permitted.

void fmpz_lucas_chain_double(fmpz_t U2m, fmpz_t U2m1, const fmpz_t Um, const fmpz_t Um1, const fmpz_t A, const fmpz_t B, const fmpz_t n)

  Given \(U_m, U_{m + 1} \pmod{n}\) compute \(U_{2m}, U_{2m + 1} \pmod{n}\).

  Aliasing of \(U_{2m}\) and \(U_m\) and aliasing of \(U_{2m + 1}\) and \(U_{m + 1}\) is permitted. No other aliasing is allowed.

void fmpz_lucas_chain_add(fmpz_t Umn, fmpz_t Umn1, const fmpz_t Um, const fmpz_t Um1, const fmpz_t Un, const fmpz_t Un1, const fmpz_t A, const fmpz_t B, const fmpz_t n)

  Given \(U_m, U_{m + 1} \pmod{n}\) and \(U_n, U_{n + 1} \pmod{n}\) compute \(U_{m + n}, U_{m + n + 1} \pmod{n}\).

  Aliasing of \(U_{m + n}\) with \(U_m\) or \(U_n\) and aliasing of \(U_{m + n + 1}\) with \(U_{m + 1}\) or \(U_{n + 1}\) is permitted. No other aliasing is allowed.

void fmpz_lucas_chain_mul(fmpz_t Ukm, fmpz_t Ukm1, const fmpz_t Um, const fmpz_t Um1, const fmpz_t A, const fmpz_t B, const fmpz_t k, const fmpz_t n)

  Given \(U_m, U_{m + 1} \pmod{n}\) compute \(U_{km}, U_{km + 1} \pmod{n}\).

  Aliasing of \(U_{km}\) and \(U_m\) and aliasing of \(U_{km + 1}\) and \(U_{m + 1}\) is permitted. No other aliasing is allowed.

void fmpz_lucas_chain_VtoU(fmpz_t Um, fmpz_t Um1, const fmpz_t Vm, const fmpz_t Vm1, const fmpz_t A, const fmpz_t B, const fmpz_t Dinv, const fmpz_t n)

  Given \(V_m, V_{m + 1} \pmod{n}\) compute \(U_m, U_{m + 1} \pmod{n}\).

  Aliasing of \(V_m\) and \(U_m\) and aliasing of \(V_{m + 1}\) and \(U_{m + 1}\) is permitted. No other aliasing is allowed.

int fmpz_divisor_in_residue_class_lenstra(fmpz_t fac, const fmpz_t n, const fmpz_t r, const fmpz_t s)

  If there exists a proper divisor of \(n\) which is \(r \pmod{s}\) for \(0 < r < s < n\), this function returns \(1\) and sets `fac` to such a divisor. Otherwise the function returns \(0\) and the value of `fac` is undefined.

  We require \(\gcd(r, s) = 1\).

  This is efficient if \(s^3 > n\).

void fmpz_nextprime(fmpz_t res, const fmpz_t n, int proved)

  Finds the next prime number larger than \(n\).

  If `proved` is nonzero, then the integer returned is guaranteed to actually be prime. Otherwise if \(n\) fits in `FLINT_BITS - 3` bits `n_nextprime` is called, and if not then the GMP `mpz_nextprime` function is called which uses a BPSW test.

## Special functions

void fmpz_primorial(fmpz_t res, ulong n)

  Sets `res` to `n` primorial or \(n \#\), the product of all prime numbers less than or equal to \(n\).

void fmpz_factor_euler_phi(fmpz_t res, const fmpz_factor_t fac)
void fmpz_euler_phi(fmpz_t res, const fmpz_t n)

  Sets `res` to the Euler totient function \(\phi(n)\), counting the number of positive integers less than or equal to \(n\) that are coprime to \(n\). The factor version takes a precomputed factorisation of \(n\).

int fmpz_factor_moebius_mu(const fmpz_factor_t fac)
int fmpz_moebius_mu(const fmpz_t n)

  Computes the Moebius function \(\mu(n)\), which is defined as \(\mu(n) = 0\) if \(n\) has a prime factor of multiplicity greater than \(1\), \(\mu(n) = -1\) if \(n\) has an odd number of distinct prime factors, and \(\mu(n) = 1\) if \(n\) has an even number of distinct prime factors. By convention, \(\mu(0) = 0\). The factor version takes a precomputed factorisation of \(n\).

void fmpz_factor_divisor_sigma(fmpz_t res, ulong k, const fmpz_factor_t fac)
void fmpz_divisor_sigma(fmpz_t res, ulong k, const fmpz_t n)

  Sets `res` to \(\sigma_k(n)\), the sum of \(k\)th powers of all divisors of \(n\). The factor version takes a precomputed factorisation of \(n\).