243 lines
8.0 KiB
C
243 lines
8.0 KiB
C
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/*============================================================================
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This file is part of FLINT.
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FLINT is free software; you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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FLINT is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with FLINT; if not, write to the Free Software
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Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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===============================================================================*/
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/******************************************************************************
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Copyright (C) 2010-2012 Fredrik Johansson
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******************************************************************************/
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#ifndef ARITH_H
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#define ARITH_H
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#include <gmp.h>
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#include <mpfr.h>
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#include "flint.h"
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#include "fmpz.h"
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#include "fmpz_mat.h"
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#include "fmpz_poly.h"
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#include "fmpq_poly.h"
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#include "fmpq.h"
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#ifdef __cplusplus
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extern "C" {
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#endif
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/* MPFR extras ***************************************************************/
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void mpfr_zeta_inv_euler_product(mpfr_t res, ulong s, int char_4);
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void mpfr_pi_chudnovsky(mpfr_t res, mpfr_rnd_t rnd);
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/* Various arithmetic functions **********************************************/
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void arith_primorial(fmpz_t res, slong n);
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void _arith_harmonic_number(fmpz_t num, fmpz_t den, slong n);
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void arith_harmonic_number(fmpq_t x, slong n);
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void arith_ramanujan_tau(fmpz_t res, const fmpz_t n);
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void arith_ramanujan_tau_series(fmpz_poly_t res, slong n);
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void arith_divisors(fmpz_poly_t res, const fmpz_t n);
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void arith_divisor_sigma(fmpz_t res, const fmpz_t n, ulong k);
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int arith_moebius_mu(const fmpz_t n);
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void arith_euler_phi(fmpz_t res, const fmpz_t n);
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/* Stirling numbers **********************************************************/
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void arith_stirling_number_1u(fmpz_t s, slong n, slong k);
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void arith_stirling_number_1(fmpz_t s, slong n, slong k);
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void arith_stirling_number_2(fmpz_t s, slong n, slong k);
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void arith_stirling_number_1u_vec(fmpz * row, slong n, slong klen);
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void arith_stirling_number_1_vec(fmpz * row, slong n, slong klen);
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void arith_stirling_number_2_vec(fmpz * row, slong n, slong klen);
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void arith_stirling_number_1u_vec_next(fmpz * row,
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const fmpz * prev, slong n, slong klen);
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void arith_stirling_number_1_vec_next(fmpz * row,
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const fmpz * prev, slong n, slong klen);
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void arith_stirling_number_2_vec_next(fmpz * row,
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const fmpz * prev, slong n, slong klen);
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void arith_stirling_matrix_1u(fmpz_mat_t mat);
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void arith_stirling_matrix_1(fmpz_mat_t mat);
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void arith_stirling_matrix_2(fmpz_mat_t mat);
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/* Bell numbers **************************************************************/
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#if FLINT64
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#define BELL_NUMBER_TAB_SIZE 26
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#else
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#define BELL_NUMBER_TAB_SIZE 16
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#endif
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extern const mp_limb_t bell_number_tab[];
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double arith_bell_number_size(ulong n);
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void arith_bell_number(fmpz_t b, ulong n);
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void arith_bell_number_bsplit(fmpz_t res, ulong n);
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void arith_bell_number_multi_mod(fmpz_t res, ulong n);
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void arith_bell_number_vec(fmpz * b, slong n);
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void arith_bell_number_vec_recursive(fmpz * b, slong n);
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void arith_bell_number_vec_multi_mod(fmpz * b, slong n);
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mp_limb_t arith_bell_number_nmod(ulong n, nmod_t mod);
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void arith_bell_number_nmod_vec(mp_ptr b, slong n, nmod_t mod);
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void arith_bell_number_nmod_vec_recursive(mp_ptr b, slong n, nmod_t mod);
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void arith_bell_number_nmod_vec_series(mp_ptr b, slong n, nmod_t mod);
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/* Euler numbers *************************************************************/
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#if FLINT64
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#define SMALL_EULER_LIMIT 25
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#else
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#define SMALL_EULER_LIMIT 15
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#endif
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static const mp_limb_t euler_number_small[] = {
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UWORD(1), UWORD(1), UWORD(5), UWORD(61), UWORD(1385), UWORD(50521), UWORD(2702765),
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UWORD(199360981),
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#if FLINT64
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UWORD(19391512145), UWORD(2404879675441), UWORD(370371188237525),
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UWORD(69348874393137901), UWORD(15514534163557086905)
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#endif
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};
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double arith_euler_number_size(ulong n);
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void arith_euler_number_vec(fmpz * res, slong n);
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void _arith_euler_number_zeta(fmpz_t res, ulong n);
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void arith_euler_number(fmpz_t res, ulong n);
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void arith_euler_polynomial(fmpq_poly_t poly, ulong n);
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/* Bernoulli numbers *********************************************************/
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#if FLINT64
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#define BERNOULLI_SMALL_NUMER_LIMIT 35
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#else
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#define BERNOULLI_SMALL_NUMER_LIMIT 27
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#endif
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static const slong _bernoulli_numer_small[] = {
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WORD(1), WORD(1), WORD(-1), WORD(1), WORD(-1), WORD(5), WORD(-691), WORD(7), WORD(-3617), WORD(43867), WORD(-174611), WORD(854513),
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WORD(-236364091), WORD(8553103),
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#if FLINT64
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WORD(-23749461029), WORD(8615841276005), WORD(-7709321041217), WORD(2577687858367)
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#endif
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};
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void _arith_bernoulli_number(fmpz_t num, fmpz_t den, ulong n);
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void arith_bernoulli_number(fmpq_t x, ulong n);
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void _arith_bernoulli_number_vec(fmpz * num, fmpz * den, slong n);
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void arith_bernoulli_number_vec(fmpq * num, slong n);
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void arith_bernoulli_number_denom(fmpz_t den, ulong n);
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double arith_bernoulli_number_size(ulong n);
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void arith_bernoulli_polynomial(fmpq_poly_t poly, ulong n);
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void _arith_bernoulli_number_zeta(fmpz_t num, fmpz_t den, ulong n);
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void _arith_bernoulli_number_vec_multi_mod(fmpz * num, fmpz * den, slong n);
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void _arith_bernoulli_number_vec_recursive(fmpz * num, fmpz * den, slong n);
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void _arith_bernoulli_number_vec_zeta(fmpz * num, fmpz * den, slong n);
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/* Cyclotomic polynomials ****************************************************/
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void _arith_cyclotomic_polynomial(fmpz * a, ulong n, mp_ptr factors,
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slong num_factors, ulong phi);
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void arith_cyclotomic_polynomial(fmpz_poly_t poly, ulong n);
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void _arith_cos_minpoly(fmpz * coeffs, slong d, ulong n);
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void arith_cos_minpoly(fmpz_poly_t poly, ulong n);
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/* Hypergeometric polynomials ************************************************/
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void arith_legendre_polynomial(fmpq_poly_t poly, ulong n);
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void arith_chebyshev_t_polynomial(fmpz_poly_t poly, ulong n);
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void arith_chebyshev_u_polynomial(fmpz_poly_t poly, ulong n);
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/* Swinnerton-Dyer polynomials ***********************************************/
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void arith_swinnerton_dyer_polynomial(fmpz_poly_t poly, ulong n);
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/* Landau function ***********************************************************/
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void arith_landau_function_vec(fmpz * res, slong len);
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/* Dedekind sums *************************************************************/
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void arith_dedekind_sum_naive(fmpq_t s, const fmpz_t h, const fmpz_t k);
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double arith_dedekind_sum_coprime_d(double h, double k);
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void arith_dedekind_sum_coprime_large(fmpq_t s, const fmpz_t h, const fmpz_t k);
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void arith_dedekind_sum_coprime(fmpq_t s, const fmpz_t h, const fmpz_t k);
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void arith_dedekind_sum(fmpq_t s, const fmpz_t h, const fmpz_t k);
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/* Exponential sums **********************************************************/
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typedef struct
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{
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int n;
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int prefactor;
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mp_limb_t sqrt_p;
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mp_limb_t sqrt_q;
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mp_limb_signed_t cos_p[FLINT_BITS];
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mp_limb_t cos_q[FLINT_BITS];
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} trig_prod_struct;
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typedef trig_prod_struct trig_prod_t[1];
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static __inline__
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void trig_prod_init(trig_prod_t sum)
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{
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sum->n = 0;
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sum->prefactor = 1;
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sum->sqrt_p = 1;
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sum->sqrt_q = 1;
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}
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void arith_hrr_expsum_factored(trig_prod_t prod, mp_limb_t k, mp_limb_t n);
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/* Number of partitions ******************************************************/
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void arith_number_of_partitions_nmod_vec(mp_ptr res, slong len, nmod_t mod);
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void arith_number_of_partitions_vec(fmpz * res, slong len);
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void arith_number_of_partitions_mpfr(mpfr_t x, ulong n);
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void arith_number_of_partitions(fmpz_t x, ulong n);
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/* Number of sums of squares representations *********************************/
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void arith_sum_of_squares(fmpz_t r, ulong k, const fmpz_t n);
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void arith_sum_of_squares_vec(fmpz * r, ulong k, slong n);
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#ifdef __cplusplus
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}
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#endif
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#endif
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