ntheory.cpp

#include <valarray>
#include <iterator>
 
#include <symengine/prime_sieve.h>
#include <symengine/ntheory.h>
#include <symengine/rational.h>
#include <symengine/add.h>
#include <symengine/mul.h>
#include <symengine/pow.h>
#ifdef HAVE_SYMENGINE_ECM
#include <ecm.h>
#endif // HAVE_SYMENGINE_ECM
#ifdef HAVE_SYMENGINE_PRIMESIEVE
#include <primesieve.hpp>
#endif // HAVE_SYMENGINE_PRIMESIEVE
#ifdef HAVE_SYMENGINE_ARB
#include <mpfr.h>
#include <flint/fmpq.h>
#include <arb.h>
#include <bernoulli.h>
#endif // HAVE_SYMENGINE_ARB
#ifndef HAVE_SYMENGINE_GMP
#include <boost/random/uniform_int.hpp>
#include <boost/random/mersenne_twister.hpp>
#include <boost/random.hpp>
#endif // !HAVE_SYMENGINE_GMP
 
namespace SymEngine
{
 
// Basic number theoretic functions
RCP<const Integer> gcd(const Integer &a, const Integer &b)
{
    integer_class g;
    mp_gcd(g, a.as_integer_class(), b.as_integer_class());
    return integer(std::move(g));
}
 
void gcd_ext(const Ptr<RCP<const Integer>> &g, const Ptr<RCP<const Integer>> &s,
             const Ptr<RCP<const Integer>> &t, const Integer &a,
             const Integer &b)
{
    integer_class g_, s_, t_;
    mp_gcdext(g_, s_, t_, a.as_integer_class(), b.as_integer_class());
    *g = integer(std::move(g_));
    *s = integer(std::move(s_));
    *t = integer(std::move(t_));
}
 
RCP<const Integer> lcm(const Integer &a, const Integer &b)
{
    integer_class c;
    mp_lcm(c, a.as_integer_class(), b.as_integer_class());
    return integer(std::move(c));
}
 
int mod_inverse(const Ptr<RCP<const Integer>> &b, const Integer &a,
                const Integer &m)
{
    int ret_val;
    integer_class inv_t;
    ret_val = mp_invert(inv_t, a.as_integer_class(), m.as_integer_class());
    *b = integer(std::move(inv_t));
    return ret_val;
}
 
RCP<const Integer> mod(const Integer &n, const Integer &d)
{
    return integer(n.as_integer_class() % d.as_integer_class());
}
 
RCP<const Integer> quotient(const Integer &n, const Integer &d)
{
    return integer(n.as_integer_class() / d.as_integer_class());
}
 
void quotient_mod(const Ptr<RCP<const Integer>> &q,
                  const Ptr<RCP<const Integer>> &r, const Integer &n,
                  const Integer &d)
{
    integer_class _q, _r;
    mp_tdiv_qr(_q, _r, n.as_integer_class(), d.as_integer_class());
    *q = integer(std::move(_q));
    *r = integer(std::move(_r));
}
 
RCP<const Integer> mod_f(const Integer &n, const Integer &d)
{
    integer_class q;
    mp_fdiv_r(q, n.as_integer_class(), d.as_integer_class());
    return integer(std::move(q));
}
 
RCP<const Integer> quotient_f(const Integer &n, const Integer &d)
{
    integer_class q;
    mp_fdiv_q(q, n.as_integer_class(), d.as_integer_class());
    return integer(std::move(q));
}
 
void quotient_mod_f(const Ptr<RCP<const Integer>> &q,
                    const Ptr<RCP<const Integer>> &r, const Integer &n,
                    const Integer &d)
{
    integer_class _q, _r;
    mp_fdiv_qr(_q, _r, n.as_integer_class(), d.as_integer_class());
    *q = integer(std::move(_q));
    *r = integer(std::move(_r));
}
 
RCP<const Integer> fibonacci(unsigned long n)
{
    integer_class f;
    mp_fib_ui(f, n);
    return integer(std::move(f));
}
 
void fibonacci2(const Ptr<RCP<const Integer>> &g,
                const Ptr<RCP<const Integer>> &s, unsigned long n)
{
    integer_class g_t;
    integer_class s_t;
    mp_fib2_ui(g_t, s_t, n);
    *g = integer(std::move(g_t));
    *s = integer(std::move(s_t));
}
 
RCP<const Integer> lucas(unsigned long n)
{
    integer_class f;
    mp_lucnum_ui(f, n);
    return integer(std::move(f));
}
 
void lucas2(const Ptr<RCP<const Integer>> &g, const Ptr<RCP<const Integer>> &s,
            unsigned long n)
{
    integer_class g_t;
    integer_class s_t;
    mp_lucnum2_ui(g_t, s_t, n);
    *g = integer(std::move(g_t));
    *s = integer(std::move(s_t));
}
 
// Binomial Coefficient
RCP<const Integer> binomial(const Integer &n, unsigned long k)
{
    integer_class f;
    mp_bin_ui(f, n.as_integer_class(), k);
    return integer(std::move(f));
}
 
// Factorial
RCP<const Integer> factorial(unsigned long n)
{
    integer_class f;
    mp_fac_ui(f, n);
    return integer(std::move(f));
}
 
// Returns true if `b` divides `a` without reminder
bool divides(const Integer &a, const Integer &b)
{
    return mp_divisible_p(a.as_integer_class(), b.as_integer_class()) != 0;
}
 
// Prime functions
int probab_prime_p(const Integer &a, unsigned reps)
{
    return mp_probab_prime_p(a.as_integer_class(), reps);
}
 
RCP<const Integer> nextprime(const Integer &a)
{
    integer_class c;
    mp_nextprime(c, a.as_integer_class());
    return integer(std::move(c));
}
 
namespace
{
// Factoring by Trial division using primes only
int _factor_trial_division_sieve(integer_class &factor, const integer_class &N)
{
    integer_class sqrtN = mp_sqrt(N);
    unsigned long limit = mp_get_ui(sqrtN);
    if (limit > std::numeric_limits<unsigned>::max())
        throw SymEngineException("N too large to factor");
    Sieve::iterator pi(numeric_cast<unsigned>(limit));
    unsigned p;
    while ((p = pi.next_prime()) <= limit) {
        if (N % p == 0) {
            factor = p;
            return 1;
        }
    }
    return 0;
}
// Factor using lehman method.
int _factor_lehman_method(integer_class &rop, const integer_class &n)
{
    if (n < 21)
        throw SymEngineException("Require n >= 21 to use lehman method");
 
    int ret_val = 0;
    integer_class u_bound;
 
    mp_root(u_bound, n, 3);
    u_bound = u_bound + 1;
 
    Sieve::iterator pi(numeric_cast<unsigned>(mp_get_ui(u_bound)));
    unsigned p;
    while ((p = pi.next_prime()) <= mp_get_ui(u_bound)) {
        if (n % p == 0) {
            rop = n / p;
            ret_val = 1;
            break;
        }
    }
 
    if (not ret_val) {
 
        integer_class k, a, b, l;
 
        k = 1;
 
        while (k <= u_bound) {
            a = mp_sqrt(4 * k * n);
            mp_root(b, n, 6);
            mp_root(l, k, 2);
            b = b / (4 * l);
            b = b + a;
 
            while (a <= b) {
                l = a * a - 4 * k * n;
                if (mp_perfect_square_p(l)) {
                    b = a + mp_sqrt(l);
                    mp_gcd(rop, n, b);
                    ret_val = 1;
                    break;
                }
                a = a + 1;
            }
            if (ret_val)
                break;
            k = k + 1;
        }
    }
 
    return ret_val;
}
} // anonymous namespace
 
int factor_lehman_method(const Ptr<RCP<const Integer>> &f, const Integer &n)
{
    int ret_val;
    integer_class rop;
 
    ret_val = _factor_lehman_method(rop, n.as_integer_class());
    *f = integer(std::move(rop));
    return ret_val;
}
 
namespace
{
// Factor using Pollard's p-1 method
int _factor_pollard_pm1_method(integer_class &rop, const integer_class &n,
                               const integer_class &c, unsigned B)
{
    if (n < 4 or B < 3)
        throw SymEngineException(
            "Require n > 3 and B > 2 to use Pollard's p-1 method");
 
    integer_class m, _c;
    _c = c;
 
    Sieve::iterator pi(B);
    unsigned p;
    while ((p = pi.next_prime()) <= B) {
        m = 1;
        // calculate log(p, B), this can be improved
        while (m <= B / p) {
            m = m * p;
        }
        mp_powm(_c, _c, m, n);
    }
    _c = _c - 1;
    mp_gcd(rop, _c, n);
 
    if (rop == 1 or rop == n)
        return 0;
    else
        return 1;
}
} // anonymous namespace
 
int factor_pollard_pm1_method(const Ptr<RCP<const Integer>> &f,
                              const Integer &n, unsigned B, unsigned retries)
{
    int ret_val = 0;
    integer_class rop, nm4, c;
 
    mp_randstate state;
    nm4 = n.as_integer_class() - 4;
 
    for (unsigned i = 0; i < retries and ret_val == 0; ++i) {
        state.urandomint(c, nm4);
        c += 2;
        ret_val = _factor_pollard_pm1_method(rop, n.as_integer_class(), c, B);
    }
 
    if (ret_val != 0)
        *f = integer(std::move(rop));
    return ret_val;
}
 
namespace
{
// Factor using Pollard's rho method
int _factor_pollard_rho_method(integer_class &rop, const integer_class &n,
                               const integer_class &a, const integer_class &s,
                               unsigned steps = 10000)
{
    if (n < 5)
        throw SymEngineException("Require n > 4 to use pollard's-rho method");
 
    integer_class u, v, g, m;
    u = s;
    v = s;
 
    for (unsigned i = 0; i < steps; ++i) {
        u = (u * u + a) % n;
        v = (v * v + a) % n;
        v = (v * v + a) % n;
        m = u - v;
        mp_gcd(g, m, n);
 
        if (g == n)
            return 0;
        if (g == 1)
            continue;
        rop = g;
        return 1;
    }
    return 0;
}
} // namespace
 
int factor_pollard_rho_method(const Ptr<RCP<const Integer>> &f,
                              const Integer &n, unsigned retries)
{
    int ret_val = 0;
    integer_class rop, nm1, nm4, a, s;
    mp_randstate state;
    nm1 = n.as_integer_class() - 1;
    nm4 = n.as_integer_class() - 4;
 
    for (unsigned i = 0; i < retries and ret_val == 0; ++i) {
        state.urandomint(a, nm1);
        state.urandomint(s, nm4);
        s += 1;
        ret_val = _factor_pollard_rho_method(rop, n.as_integer_class(), a, s);
    }
 
    if (ret_val != 0)
        *f = integer(std::move(rop));
    return ret_val;
}
 
// Factorization
int factor(const Ptr<RCP<const Integer>> &f, const Integer &n, double B1)
{
    int ret_val = 0;
    integer_class _n, _f;
 
    _n = n.as_integer_class();
 
#ifdef HAVE_SYMENGINE_ECM
    if (mp_perfect_power_p(_n)) {
 
        unsigned long int i = 1;
        integer_class m, rem;
        rem = 1; // Any non zero number
        m = 2;   // set `m` to 2**i, i = 1 at the begining
 
        // calculate log2n, this can be improved
        for (; m < _n; ++i)
            m = m * 2;
 
        // eventually `rem` = 0 zero as `n` is a perfect power. `f_t` will
        // be set to a factor of `n` when that happens
        while (i > 1 and rem != 0) {
            mp_rootrem(_f, rem, _n, i);
            --i;
        }
 
        ret_val = 1;
    } else {
 
        if (mp_probab_prime_p(_n, 25) > 0) { // most probably, n is a prime
            ret_val = 0;
            _f = _n;
        } else {
 
            for (int i = 0; i < 10 and not ret_val; ++i)
                ret_val = ecm_factor(get_mpz_t(_f), get_mpz_t(_n), B1, nullptr);
            mp_demote(_f);
            if (not ret_val)
                throw SymEngineException(
                    "ECM failed to factor the given number");
        }
    }
#else
    // B1 is discarded if gmp-ecm is not installed
    ret_val = _factor_trial_division_sieve(_f, _n);
#endif // HAVE_SYMENGINE_ECM
    *f = integer(std::move(_f));
 
    return ret_val;
}
 
int factor_trial_division(const Ptr<RCP<const Integer>> &f, const Integer &n)
{
    int ret_val;
    integer_class factor;
    ret_val = _factor_trial_division_sieve(factor, n.as_integer_class());
    if (ret_val == 1)
        *f = integer(std::move(factor));
    return ret_val;
}
 
void prime_factors(std::vector<RCP<const Integer>> &prime_list,
                   const Integer &n)
{
    integer_class sqrtN;
    integer_class _n = n.as_integer_class();
    if (_n == 0)
        return;
    if (_n < 0)
        _n *= -1;
 
    sqrtN = mp_sqrt(_n);
    auto limit = mp_get_ui(sqrtN);
    if (not mp_fits_ulong_p(sqrtN)
        or limit > std::numeric_limits<unsigned>::max())
        throw SymEngineException("N too large to factor");
    Sieve::iterator pi(numeric_cast<unsigned>(limit));
    unsigned p;
 
    while ((p = pi.next_prime()) <= limit) {
        while (_n % p == 0) {
            prime_list.push_back(integer(p));
            _n = _n / p;
        }
        if (_n == 1)
            break;
    }
    if (not(_n == 1))
        prime_list.push_back(integer(std::move(_n)));
}
 
void prime_factor_multiplicities(map_integer_uint &primes_mul, const Integer &n)
{
    integer_class sqrtN;
    integer_class _n = n.as_integer_class();
    unsigned count;
    if (_n == 0)
        return;
    if (_n < 0)
        _n *= -1;
 
    sqrtN = mp_sqrt(_n);
    auto limit = mp_get_ui(sqrtN);
    if (not mp_fits_ulong_p(sqrtN)
        or limit > std::numeric_limits<unsigned>::max())
        throw SymEngineException("N too large to factor");
    Sieve::iterator pi(numeric_cast<unsigned>(limit));
 
    unsigned p;
    while ((p = pi.next_prime()) <= limit) {
        count = 0;
        while (_n % p == 0) { // when a prime factor is found, we divide
            ++count;          // _n by that prime as much as we can
            _n = _n / p;
        }
        if (count > 0) {
            insert(primes_mul, integer(p), count);
            if (_n == 1)
                break;
        }
    }
    if (not(_n == 1))
        insert(primes_mul, integer(std::move(_n)), 1);
}
 
RCP<const Number> bernoulli(unsigned long n)
{
#ifdef HAVE_SYMENGINE_ARB
    fmpq_t res;
    fmpq_init(res);
    bernoulli_fmpq_ui(res, n);
    mpq_t a;
    mpq_init(a);
    fmpq_get_mpq(a, res);
    rational_class b(a);
    fmpq_clear(res);
    mpq_clear(a);
    return Rational::from_mpq(std::move(b));
#else
    // TODO: implement a faster algorithm
    std::vector<rational_class> v(n + 1);
    for (unsigned m = 0; m <= n; ++m) {
        v[m] = rational_class(1u, m + 1);
 
        for (unsigned j = m; j >= 1; --j) {
            v[j - 1] = j * (v[j - 1] - v[j]);
        }
    }
    return Rational::from_mpq(v[0]);
#endif
}
 
RCP<const Number> harmonic(unsigned long n, long m)
{
    rational_class res(0);
    if (m == 1) {
        for (unsigned i = 1; i <= n; ++i) {
            res += rational_class(1u, i);
        }
        return Rational::from_mpq(res);
    } else {
        for (unsigned i = 1; i <= n; ++i) {
            if (m > 0) {
                rational_class t(1u, i);
#if SYMENGINE_INTEGER_CLASS != SYMENGINE_BOOSTMP
                mp_pow_ui(get_den(t), get_den(t), m);
#else
                mp_pow_ui(t, t, m);
#endif
                res += t;
            } else {
                integer_class t(i);
                mp_pow_ui(t, t, static_cast<unsigned long>(-m));
                res += t;
            }
        }
        return Rational::from_mpq(res);
    }
}
 
// References : Cohen H., A course in computational algebraic number theory
// (1996), page 21.
bool crt(const Ptr<RCP<const Integer>> &R,
         const std::vector<RCP<const Integer>> &rem,
         const std::vector<RCP<const Integer>> &mod)
{
    if (mod.size() > rem.size())
        throw SymEngineException("Too few remainders");
    if (mod.size() == 0)
        throw SymEngineException("Moduli vector cannot be empty");
 
    integer_class m, r, g, s, t;
    m = mod[0]->as_integer_class();
    r = rem[0]->as_integer_class();
 
    for (unsigned i = 1; i < mod.size(); ++i) {
        mp_gcdext(g, s, t, m, mod[i]->as_integer_class());
        // g = s * m + t * mod[i]
        t = rem[i]->as_integer_class() - r;
        if (not mp_divisible_p(t, g))
            return false;
        r += m * s * (t / g); // r += m * (m**-1 mod[i]/g)* (rem[i] - r) / g
        m *= mod[i]->as_integer_class() / g;
        mp_fdiv_r(r, r, m);
    }
    *R = integer(std::move(r));
    return true;
}
 
namespace
{
// Crt over a cartesian product of vectors (Assuming that moduli are pairwise
// relatively prime).
void _crt_cartesian(std::vector<RCP<const Integer>> &R,
                    const std::vector<std::vector<RCP<const Integer>>> &rem,
                    const std::vector<RCP<const Integer>> &mod)
{
    if (mod.size() > rem.size())
        throw SymEngineException("Too few remainders");
    if (mod.size() == 0)
        throw SymEngineException("Moduli vector cannot be empty");
    integer_class m, _m, r, s, t;
    m = mod[0]->as_integer_class();
    R = rem[0];
 
    for (unsigned i = 1; i < mod.size(); ++i) {
        std::vector<RCP<const Integer>> rem2;
        mp_invert(s, m, mod[i]->as_integer_class());
        _m = m;
        m *= mod[i]->as_integer_class();
        for (auto &elem : R) {
            for (auto &_k : rem[i]) {
                r = elem->as_integer_class();
                r += _m * s * (_k->as_integer_class() - r);
                mp_fdiv_r(r, r, m);
                rem2.push_back(integer(r));
            }
        }
        R = rem2;
    }
}
 
// Tests whether n is a prime power and finds a prime p and e such that n =
// p**e.
bool _prime_power(integer_class &p, integer_class &e, const integer_class &n)
{
    if (n < 2)
        return false;
    integer_class _n = n, temp;
    e = 1;
    unsigned i = 2;
    while (mp_perfect_power_p(_n) and _n >= 2) {
        if (mp_root(temp, _n, i)) {
            e *= i;
            _n = temp;
        } else {
            ++i;
        }
    }
    if (mp_probab_prime_p(_n, 25)) {
        p = _n;
        return true;
    }
    return false;
}
 
// Computes a primitive root modulo p**e or 2*p**e where p is an odd prime.
// References : Cohen H., A course in computational algebraic number theory
// (2009), pages 25-27.
void _primitive_root(integer_class &g, const integer_class &p,
                     const integer_class &e, bool even = false)
{
    std::vector<RCP<const Integer>> primes;
    prime_factors(primes, *integer(p - 1));
 
    integer_class t;
    g = 2;
    while (g < p) {
        bool root = true;
        for (const auto &it : primes) {
            t = it->as_integer_class();
            t = (p - 1) / t;
            mp_powm(t, g, t, p);
            if (t == 1) { // If g**(p-1)/q is 1 then g is not a primitive root.
                root = false;
                break;
            }
        }
        if (root)
            break;
        ++g;
    }
 
    if (e > 1) {
        t = p * p;
        integer_class pm1 = p - 1;
        mp_powm(t, g, pm1, t);
        if (t == 1) { // If g**(p-1) mod (p**2) == 1 then g + p is a primitive
                      // root.
            g += p;
        }
    }
    if (even and g % 2 == 0) {
        mp_pow_ui(t, p, mp_get_ui(e));
        g += t; // If g is even then root of 2*p**e is g + p**e.
    }
}
 
} // anonymous namespace
 
bool primitive_root(const Ptr<RCP<const Integer>> &g, const Integer &n)
{
    integer_class _n = n.as_integer_class();
    if (_n < 0)
        _n = -_n;
    if (_n <= 1)
        return false;
    if (_n < 5) {
        *g = integer(_n - 1);
        return true;
    }
    bool even = false;
    if (_n % 2 == 0) {
        if (_n % 4 == 0) {
            return false; // If n mod 4 == 0 and n > 4, then no primitive roots.
        }
        _n /= 2;
        even = true;
    }
    integer_class p, e;
    if (not _prime_power(p, e, _n))
        return false;
    _primitive_root(_n, p, e, even);
    *g = integer(std::move(_n));
    return true;
}
 
namespace
{
// Computes primitive roots modulo p**e or 2*p**e where p is an odd prime.
// References :
// [1] Cohen H., A course in computational algebraic number theory (1996), pages
// 25-27.
// [2] Hackman P., Elementary number theory (2009), page 28.
void _primitive_root_list(std::vector<RCP<const Integer>> &roots,
                          const integer_class &p, const integer_class &e,
                          bool even = false)
{
    integer_class g, h, d, t, pe2, n, pm1;
    _primitive_root(g, p, integer_class(1),
                    false); // Find one primitive root for p.
    h = 1;
    pm1 = p - 1;
    // Generate other primitive roots for p. h = g**i and gcd(i, p-1) = 1.
    // Ref[2]
    mp_pow_ui(n, p, mp_get_ui(e));
    for (unsigned long i = 1; i < p; ++i) {
        h *= g;
        h %= p;
        mp_gcd(d, pm1, integer_class(i));
        if (d == 1) {
            if (e == 1) {
                if (even and h % 2 == 0)
                    roots.push_back(integer(h + n));
                else
                    roots.push_back(integer(h));
            } else {
                integer_class pp = p * p;
                // Find d such that (h + d*p)**(p-1) mod (p**2) == 1. Ref[1]
                // h**(p-1) - 1 = d*p*h**(p-2)
                // d = (h - h**(2-p)) / p
                t = 2 - p;
                mp_powm(d, h, t, pp);
                d = ((h - d) / p + p) % p;
                t = h;
                // t = h + i * p + j * p * p and i != d
                mp_pow_ui(pe2, p, mp_get_ui(e) - 2);
                for (unsigned long j = 0; j < pe2; ++j) {
                    for (unsigned long i = 0; i < p; ++i) {
                        if (i != d) {
                            if (even and t % 2 == 0)
                                roots.push_back(integer(t + n));
                            else
                                roots.push_back(integer(t));
                        }
                        t += p;
                    }
                }
            }
        }
    }
} //_primitive_root_list
} // anonymous namespace
 
void primitive_root_list(std::vector<RCP<const Integer>> &roots,
                         const Integer &n)
{
    integer_class _n = n.as_integer_class();
    if (_n < 0)
        _n = -_n;
    if (_n <= 1)
        return;
    if (_n < 5) {
        roots.push_back(integer(_n - 1));
        return;
    }
    bool even = false;
    if (_n % 2 == 0) {
        if (_n % 4 == 0) {
            return; // If n%4 == 0 and n > 4, then no primitive roots.
        }
        _n /= 2;
        even = true;
    }
    integer_class p, e;
    if (not _prime_power(p, e, _n))
        return;
    _primitive_root_list(roots, p, e, even);
    std::sort(roots.begin(), roots.end(), SymEngine::RCPIntegerKeyLess());
    return;
}
 
RCP<const Integer> totient(const RCP<const Integer> &n)
{
    if (n->is_zero())
        return integer(1);
 
    integer_class phi = n->as_integer_class(), p;
    if (phi < 0)
        phi = -phi;
    map_integer_uint prime_mul;
    prime_factor_multiplicities(prime_mul, *n);
 
    for (const auto &it : prime_mul) {
        p = it.first->as_integer_class();
        mp_divexact(phi, phi, p);
        // phi is exactly divisible by p.
        phi *= p - 1;
    }
    return integer(std::move(phi));
}
 
RCP<const Integer> carmichael(const RCP<const Integer> &n)
{
    if (n->is_zero())
        return integer(1);
 
    map_integer_uint prime_mul;
    integer_class lambda, t, p;
    unsigned multiplicity;
 
    prime_factor_multiplicities(prime_mul, *n);
    lambda = 1;
    for (const auto &it : prime_mul) {
        p = it.first->as_integer_class();
        multiplicity = it.second;
        if (p == 2
            and multiplicity
                    > 2) { // For powers of 2 greater than 4 divide by 2.
            multiplicity--;
        }
        t = p - 1;
        mp_lcm(lambda, lambda, t);
        mp_pow_ui(t, p, multiplicity - 1);
        // lambda and p are relatively prime.
        lambda = lambda * t;
    }
    return integer(std::move(lambda));
}
 
// References : Cohen H., A course in computational algebraic number theory
// (1996), page 25.
bool multiplicative_order(const Ptr<RCP<const Integer>> &o,
                          const RCP<const Integer> &a,
                          const RCP<const Integer> &n)
{
    integer_class order, p, t;
    integer_class _a = a->as_integer_class(),
                  _n = mp_abs(n->as_integer_class());
    mp_gcd(t, _a, _n);
    if (t != 1)
        return false;
 
    RCP<const Integer> lambda = carmichael(n);
    map_integer_uint prime_mul;
    prime_factor_multiplicities(prime_mul, *lambda);
    _a %= _n;
    order = lambda->as_integer_class();
 
    for (const auto &it : prime_mul) {
        p = it.first->as_integer_class();
        mp_pow_ui(t, p, it.second);
        mp_divexact(order, order, t);
        mp_powm(t, _a, order, _n);
        while (t != 1) {
            mp_powm(t, t, p, _n);
            order *= p;
        }
    }
    *o = integer(std::move(order));
    return true;
}
int legendre(const Integer &a, const Integer &n)
{
    return mp_legendre(a.as_integer_class(), n.as_integer_class());
}
 
int jacobi(const Integer &a, const Integer &n)
{
    return mp_jacobi(a.as_integer_class(), n.as_integer_class());
}
 
int kronecker(const Integer &a, const Integer &n)
{
    return mp_kronecker(a.as_integer_class(), n.as_integer_class());
}
 
namespace
{
bool _sqrt_mod_tonelli_shanks(integer_class &rop, const integer_class &a,
                              const integer_class &p)
{
    mp_randstate state;
    integer_class n, y, b, q, pm1, t(1);
    pm1 = p - 1;
    unsigned e, m;
    e = numeric_cast<unsigned>(mp_scan1(pm1));
    q = pm1 >> e; // p - 1 = 2**e*q
 
    while (t != -1) {
        state.urandomint(n, p);
        t = mp_legendre(n, p);
    }
    mp_powm(y, n, q, p); // y = n**q mod p
    mp_powm(b, a, q, p); // b = a**q mod p
    t = (q + 1) / 2;
    mp_powm(rop, a, t, p); // rop = a**((q + 1) / 2) mod p
 
    while (b != 1) {
        m = 0;
        t = b;
        while (t != 1) {
            mp_powm(t, t, integer_class(2), p);
            ++m; // t = t**2 = b**2**(m)
        }
        if (m == e)
            return false;
        mp_pow_ui(q, integer_class(2), e - m - 1); // q = 2**(e - m - 1)
        mp_powm(t, y, q, p);                       // t = y**(2**(e - m - 1))
        mp_powm(y, t, integer_class(2), p);        // y = t**2
        e = m;
        rop = (rop * t) % p;
        b = (b * y) % p;
    }
    return true;
}
 
bool _sqrt_mod_prime(integer_class &rop, const integer_class &a,
                     const integer_class &p)
{
    if (p == 2) {
        rop = a % p;
        return true;
    }
    int l = mp_legendre(a, p);
    integer_class t;
    if (l == -1) {
        return false;
    } else if (l == 0) {
        rop = 0;
    } else if (p % 4 == 3) {
        t = (p + 1) / 4;
        mp_powm(rop, a, t, p);
    } else if (p % 8 == 5) {
        t = (p - 1) / 4;
        mp_powm(t, a, t, p);
        if (t == 1) {
            t = (p + 3) / 8;
            mp_powm(rop, a, t, p);
        } else {
            t = (p - 5) / 8;
            integer_class t1 = 4 * a;
            mp_powm(t, t1, t, p);
            rop = (2 * a * t) % p;
        }
    } else {
        if (p < 10000) { // If p < 10000, brute force is faster.
            integer_class sq = integer_class(1), _a;
            mp_fdiv_r(_a, a, p);
            for (unsigned i = 1; i < p; ++i) {
                if (sq == _a) {
                    rop = i;
                    return true;
                }
                sq += 2 * i + 1;
                mp_fdiv_r(sq, sq, p);
            }
            return false;
        } else {
            return _sqrt_mod_tonelli_shanks(rop, a, p);
        }
    }
    return true;
}
 
// References : Menezes, Alfred J., Paul C. Van Oorschot, and Scott A. Vanstone.
// Handbook of applied cryptography. CRC press, 2010. pages 104 - 108
// Calculates log = x mod q**k where g**x == a mod p and order(g, p) = n.
void _discrete_log(integer_class &log, const integer_class &a,
                   const integer_class &g, const integer_class &n,
                   const integer_class &q, const unsigned &k,
                   const integer_class &p)
{
    log = 0;
    integer_class gamma = a, alpha, _n, t, beta, qj(1), m, l;
    _n = n / q;
    mp_powm(alpha, g, _n, p);
    mp_sqrtrem(m, t, q);
    if (t != 0)
        ++m; // m = ceiling(sqrt(q)).
    map_integer_uint
        table; // Table for lookup in baby-step giant-step algorithm
    integer_class alpha_j(1), d, s;
    s = -m;
    mp_powm(s, alpha, s, p);
 
    for (unsigned j = 0; j < m; ++j) {
        insert(table, integer(alpha_j), j);
        alpha_j = (alpha_j * alpha) % p;
    }
 
    for (unsigned long j = 0; j < k; ++j) { // Pohlig-Hellman
        mp_powm(beta, gamma, _n, p);
        // Baby-step giant-step algorithm for l = log_alpha(beta)
        d = beta;
        bool found = false;
        for (unsigned i = 0; not found && i < m; ++i) {
            if (table.find(integer(d)) != table.end()) {
                l = i * m + table[integer(d)];
                found = true;
                break;
            }
            d = (d * s) % p;
        }
        _n /= q;
        t = -l * qj;
 
        log -= t;
        mp_powm(t, g, t, p);
        gamma *= t; // gamma *= g ** (-l * (q ** j))
        qj *= q;
    }
}
 
// References : Johnston A., A generalised qth root algorithm.
// Solution for x**n == a mod p**k where a != 0 mod p and p is an odd prime.
bool _nthroot_mod1(std::vector<RCP<const Integer>> &roots,
                   const integer_class &a, const integer_class &n,
                   const integer_class &p, const unsigned k,
                   bool all_roots = false)
{
    integer_class _n, r, root, s, t, g(0), pk, m, phi;
    mp_pow_ui(pk, p, k);
    phi = pk * (p - 1) / p;
    mp_gcd(m, phi, n);
    t = phi / m;
    mp_powm(t, a, t, pk);
    // Check whether a**(phi / gcd(phi, n)) == 1 mod p**k.
    if (t != 1) {
        return false;
    }
    // Solve x**n == a mod p first.
    t = p - 1;
    mp_gcdext(_n, r, s, n, t);
    if (r < 0) {
        mp_fdiv_r(r, r, t / _n);
    }
    mp_powm(s, a, r, p);
 
    // Solve x**(_n) == s mod p where _n | p - 1.
    if (_n == 1) {
        root = s;
    } else if (_n == 2) {
        _sqrt_mod_prime(root, s, p);
    } else { // Ref[1]
        map_integer_uint prime_mul;
        prime_factor_multiplicities(prime_mul, *integer(_n));
        integer_class h, q, qt, z, v, x, s1 = s;
        _primitive_root(g, p, integer_class(2));
        unsigned c;
        for (const auto &it : prime_mul) {
            q = it.first->as_integer_class();
            mp_pow_ui(qt, q, it.second);
            h = (p - 1) / q;
            c = 1;
            while (h % q == 0) {
                ++c;
                h /= q;
            }
            mp_invert(t, h, qt);
            z = t * -h;
            x = (1 + z) / qt;
            mp_powm(v, s1, x, p);
 
            if (c == it.second) {
                s1 = v;
            } else {
                mp_powm(x, s1, h, p);
                t = h * qt;
                mp_powm(r, g, t, p);
                mp_pow_ui(qt, q, c - it.second);
                _discrete_log(t, x, r, qt, q, c - it.second, p);
                t = -z * t;
                mp_powm(r, g, t, p);
                v *= r;
                mp_fdiv_r(v, v, p);
                s1 = v;
            }
        }
        root = s1;
    }
    r = n;
    unsigned c = 0;
    while (r % p == 0) {
        mp_divexact(r, r, p);
        ++c;
    }
 
    // Solve s == x**r mod p**k where (x**r)**(p**c)) == a mod p**k
    integer_class pc = n / r, pd = pc * p;
    if (c >= 1) {
        mp_powm(s, root, r, p);
        // s == root**r mod p. Since s**(p**c) == 1 == a mod p**(c + 1), lift
        // until p**k.
        for (unsigned d = c + 2; d <= k; ++d) {
            t = 1 - pc;
            pd *= p;
            mp_powm(t, s, t, pd);
            t = (a * t - s) / pc;
            s += t;
        }
    } else {
        s = a;
    }
 
    // Solve x**r == s mod p**k given that root**r == s mod p and r % p != 0.
    integer_class u;
    pd = p;
    for (unsigned d = 2; d < 2 * k; d *= 2) { // Hensel lifting
        t = r - 1;
        pd *= pd;
        if (d > k)
            pd = pk;
        mp_powm(u, root, t, pd);
        t = r * u;
        mp_invert(t, t, pd);
        root += (s - u * root) * t;
        mp_fdiv_r(root, root, pd);
    }
    if (m != 1 and all_roots) {
        // All roots are generated by root*(g**(phi / gcd(phi , n)))**j
        if (n == 2) {
            t = -1;
        } else {
            if (g == 0)
                _primitive_root(g, p, integer_class(2));
            t = phi / m;
            mp_powm(t, g, t, pk);
        }
        for (unsigned j = 0; j < m; ++j) {
            roots.push_back(integer(root));
            root *= t;
            mp_fdiv_r(root, root, pk);
        }
    } else {
        roots.push_back(integer(root));
    }
    return true;
}
 
// Checks if Solution for x**n == a mod p**k exists where a != 0 mod p and p is
// an odd prime.
bool _is_nthroot_mod1(const integer_class &a, const integer_class &n,
                      const integer_class &p, const unsigned k)
{
    integer_class t, pk, m, phi;
    mp_pow_ui(pk, p, k);
    phi = pk * (p - 1) / p;
    mp_gcd(m, phi, n);
    t = phi / m;
    mp_powm(t, a, t, pk);
    // Check whether a**(phi / gcd(phi, n)) == 1 mod p**k.
    if (t != 1) {
        return false;
    }
    return true;
}
 
// Solution for x**n == a mod p**k.
bool _nthroot_mod_prime_power(std::vector<RCP<const Integer>> &roots,
                              const integer_class &a, const integer_class &n,
                              const integer_class &p, const unsigned k,
                              bool all_roots = false)
{
    integer_class pk, root;
    std::vector<RCP<const Integer>> _roots;
    if (a % p != 0) {
        if (p == 2) {
            integer_class r = n, t, s, pc, pj;
            pk = integer_class(1) << k;
            unsigned c = numeric_cast<unsigned>(mp_scan1(n));
            r = n >> c; // n = 2**c * r where r is odd.
 
            // Handle special cases of k = 1 and k = 2.
            if (k == 1) {
                roots.push_back(integer(1));
                return true;
            }
            if (k == 2) {
                if (c > 0 and a % 4 == 3) {
                    return false;
                }
                roots.push_back(integer(a % 4));
                if (all_roots and c > 0)
                    roots.push_back(integer(3));
                return true;
            }
            if (c >= k - 2) {
                c = k - 2; // Since x**(2**c) == x**(2**(k - 2)) mod 2**k, let c
                           // = k - 2.
            }
            t = integer_class(1) << (k - 2);
            pc = integer_class(1) << c;
 
            mp_invert(s, r, t);
            if (c == 0) {
                // x**r == a mod 2**k and x**2**(k - 2) == 1 mod 2**k, implies
                // x**(r * s) == x == a**s mod 2**k.
                mp_powm(root, a, s, pk);
                roots.push_back(integer(root));
                return true;
            }
 
            // First, solve for y**2**c == a mod 2**k where y == x**r
            t = integer_class(1) << (c + 2);
            mp_fdiv_r(t, a, t);
            // Check for a == y**2**c == 1 mod 2**(c + 2).
            if (t != 1)
                return false;
            root = 1;
            pj = pc * 4;
            // 1 is a root of x**2**c == 1 mod 2**(c + 2). Lift till 2**k.
            for (unsigned j = c + 2; j < k; ++j) {
                pj *= 2;
                mp_powm(t, root, pc, pj);
                t -= a;
                if (t % pj != 0)
                    // Add 2**(j - c).
                    root += integer_class(1) << (j - c);
            }
            // Solve x**r == root mod 2**k.
            mp_powm(root, root, s, pk);
 
            if (all_roots) {
                // All roots are generated by, root * (j * (2**(k - c) +/- 1)).
                t = pk / pc * root;
                for (unsigned i = 0; i < 2; ++i) {
                    for (unsigned long j = 0; j < pc; ++j) {
                        roots.push_back(integer(root));
                        root += t;
                    }
                    root = t - root;
                }
            } else {
                roots.push_back(integer(root));
            }
            return true;
        } else {
            return _nthroot_mod1(roots, a, n, p, k, all_roots);
        }
    } else {
        integer_class _a;
        mp_pow_ui(pk, p, k);
        _a = a % pk;
        unsigned m;
        integer_class pm;
        if (_a == 0) {
            if (not all_roots) {
                roots.push_back(integer(0));
                return true;
            }
            _roots.push_back(integer(0));
            if (n >= k)
                m = k - 1;
            else
                m = numeric_cast<unsigned>(k - 1 - (k - 1) / mp_get_ui(n));
            mp_pow_ui(pm, p, m);
        } else {
            unsigned r = 1;
            mp_divexact(_a, _a, p);
            while (_a % p == 0) {
                mp_divexact(_a, _a, p);
                ++r;
            }
            if (r < n or r % n != 0
                or not _nthroot_mod_prime_power(_roots, _a, n, p, k - r,
                                                all_roots)) {
                return false;
            }
            m = numeric_cast<unsigned>(r / mp_get_ui(n));
            mp_pow_ui(pm, p, m);
            if (not all_roots) {
                roots.push_back(
                    integer(_roots.back()->as_integer_class() * pm));
                return true;
            }
            for (auto &it : _roots) {
                it = integer(it->as_integer_class() * pm);
            }
            m = numeric_cast<unsigned>(r - r / mp_get_ui(n));
            mp_pow_ui(pm, p, m);
        }
        integer_class pkm;
        mp_pow_ui(pkm, p, k - m);
 
        for (const auto &it : _roots) {
            root = it->as_integer_class();
            for (unsigned long i = 0; i < pm; ++i) {
                roots.push_back(integer(root));
                root += pkm;
            }
        }
    }
    return true;
}
} // anonymous namespace
 
// Returns whether Solution for x**n == a mod p**k exists or not
bool _is_nthroot_mod_prime_power(const integer_class &a, const integer_class &n,
                                 const integer_class &p, const unsigned k)
{
    integer_class pk;
    if (a % p != 0) {
        if (p == 2) {
            integer_class t;
            unsigned c = numeric_cast<unsigned>(mp_scan1(n));
 
            // Handle special cases of k = 1 and k = 2.
            if (k == 1) {
                return true;
            }
            if (k == 2) {
                if (c > 0 and a % 4 == 3) {
                    return false;
                }
                return true;
            }
            if (c >= k - 2) {
                c = k - 2; // Since x**(2**c) == x**(2**(k - 2)) mod 2**k, let c
                           // = k - 2.
            }
            if (c == 0) {
                // x**r == a mod 2**k and x**2**(k - 2) == 1 mod 2**k, implies
                // x**(r * s) == x == a**s mod 2**k.
                return true;
            }
 
            // First, solve for y**2**c == a mod 2**k where y == x**r
            t = integer_class(1) << (c + 2);
            mp_fdiv_r(t, a, t);
            // Check for a == y**2**c == 1 mod 2**(c + 2).
            if (t != 1)
                return false;
            return true;
        } else {
            return _is_nthroot_mod1(a, n, p, k);
        }
    } else {
        integer_class _a;
        mp_pow_ui(pk, p, k);
        _a = a % pk;
        integer_class pm;
        if (_a == 0) {
            return true;
        } else {
            unsigned r = 1;
            mp_divexact(_a, _a, p);
            while (_a % p == 0) {
                mp_divexact(_a, _a, p);
                ++r;
            }
            if (r < n or r % n != 0
                or not _is_nthroot_mod_prime_power(_a, n, p, k - r)) {
                return false;
            }
            return true;
        }
    }
    return true;
}
 
bool nthroot_mod(const Ptr<RCP<const Integer>> &root,
                 const RCP<const Integer> &a, const RCP<const Integer> &n,
                 const RCP<const Integer> &mod)
{
    if (mod->as_integer_class() <= 0) {
        return false;
    } else if (mod->as_integer_class() == 1) {
        *root = integer(0);
        return true;
    }
    map_integer_uint prime_mul;
    prime_factor_multiplicities(prime_mul, *mod);
    std::vector<RCP<const Integer>> moduli;
    bool ret_val;
 
    std::vector<RCP<const Integer>> rem;
    for (const auto &it : prime_mul) {
        integer_class _mod;
        mp_pow_ui(_mod, it.first->as_integer_class(), it.second);
        moduli.push_back(integer(std::move(_mod)));
        ret_val = _nthroot_mod_prime_power(
            rem, a->as_integer_class(), n->as_integer_class(),
            it.first->as_integer_class(), it.second, false);
        if (not ret_val)
            return false;
    }
    crt(root, rem, moduli);
    return true;
}
 
void nthroot_mod_list(std::vector<RCP<const Integer>> &roots,
                      const RCP<const Integer> &a, const RCP<const Integer> &n,
                      const RCP<const Integer> &m)
{
    if (m->as_integer_class() <= 0) {
        return;
    } else if (m->as_integer_class() == 1) {
        roots.push_back(integer(0));
        return;
    }
    map_integer_uint prime_mul;
    prime_factor_multiplicities(prime_mul, *m);
    std::vector<RCP<const Integer>> moduli;
    bool ret_val;
 
    std::vector<std::vector<RCP<const Integer>>> rem;
    for (const auto &it : prime_mul) {
        integer_class _mod;
        mp_pow_ui(_mod, it.first->as_integer_class(), it.second);
        moduli.push_back(integer(std::move(_mod)));
        std::vector<RCP<const Integer>> rem1;
        ret_val = _nthroot_mod_prime_power(
            rem1, a->as_integer_class(), n->as_integer_class(),
            it.first->as_integer_class(), it.second, true);
        if (not ret_val)
            return;
        rem.push_back(rem1);
    }
    _crt_cartesian(roots, rem, moduli);
    std::sort(roots.begin(), roots.end(), SymEngine::RCPIntegerKeyLess());
}
 
bool powermod(const Ptr<RCP<const Integer>> &powm, const RCP<const Integer> &a,
              const RCP<const Number> &b, const RCP<const Integer> &m)
{
    if (is_a<Integer>(*b)) {
        integer_class t = down_cast<const Integer &>(*b).as_integer_class();
        if (b->is_negative())
            t *= -1;
        mp_powm(t, a->as_integer_class(), t, m->as_integer_class());
        if (b->is_negative()) {
            bool ret_val = mp_invert(t, t, m->as_integer_class());
            if (not ret_val)
                return false;
        }
        *powm = integer(std::move(t));
        return true;
    } else if (is_a<Rational>(*b)) {
        RCP<const Integer> num, den, r;
        get_num_den(down_cast<const Rational &>(*b), outArg(num), outArg(den));
        if (den->is_negative()) {
            den = den->mulint(*minus_one);
            num = num->mulint(*minus_one);
        }
        integer_class t = mp_abs(num->as_integer_class());
        mp_powm(t, a->as_integer_class(), t, m->as_integer_class());
        if (num->is_negative()) {
            bool ret_val = mp_invert(t, t, m->as_integer_class());
            if (not ret_val)
                return false;
        }
        r = integer(std::move(t));
        return nthroot_mod(powm, r, den, m);
    }
    return false;
}
 
void powermod_list(std::vector<RCP<const Integer>> &pows,
                   const RCP<const Integer> &a, const RCP<const Number> &b,
                   const RCP<const Integer> &m)
{
    if (is_a<Integer>(*b)) {
        integer_class t
            = mp_abs(down_cast<const Integer &>(*b).as_integer_class());
        mp_powm(t, a->as_integer_class(), t, m->as_integer_class());
        if (b->is_negative()) {
            bool ret_val = mp_invert(t, t, m->as_integer_class());
            if (not ret_val)
                return;
        }
        pows.push_back(integer(std::move(t)));
    } else if (is_a<Rational>(*b)) {
        RCP<const Integer> num, den, r;
        get_num_den(down_cast<const Rational &>(*b), outArg(num), outArg(den));
        if (den->is_negative()) {
            den = den->mulint(*integer(-1));
            num = num->mulint(*integer(-1));
        }
        integer_class t = num->as_integer_class();
        if (num->is_negative())
            t *= -1;
        mp_powm(t, a->as_integer_class(), t, m->as_integer_class());
        if (num->is_negative()) {
            bool ret_val = mp_invert(t, t, m->as_integer_class());
            if (not ret_val)
                return;
        }
        r = integer(t);
        nthroot_mod_list(pows, r, den, m);
    }
}
 
vec_integer_class quadratic_residues(const Integer &a)
{
    /*
        Returns the list of quadratic residues.
        Example
        ========
        >>> quadratic_residues(7)
        [0, 1, 2, 4]
    */
 
    if (a.as_integer_class() < 1) {
        throw SymEngineException("quadratic_residues: Input must be > 0");
    }
 
    vec_integer_class residue;
    for (integer_class i = integer_class(0); i <= a.as_int() / 2; i++) {
        residue.push_back((i * i) % a.as_int());
    }
 
    sort(residue.begin(), residue.end());
    residue.erase(unique(residue.begin(), residue.end()), residue.end());
 
    return residue;
}
 
bool is_quad_residue(const Integer &a, const Integer &p)
{
    /*
    Returns true if ``a`` (mod ``p``) is in the set of squares mod ``p``,
    i.e a % p in set([i**2 % p for i in range(p)]). If ``p`` is an odd but
    not prime, an iterative method is used to make the determination.
    */
 
    integer_class p2 = p.as_integer_class();
    if (p2 == 0)
        throw SymEngineException(
            "is_quad_residue: Second parameter must be non-zero");
    if (p2 < 0)
        p2 = -p2;
    integer_class a_final = a.as_integer_class();
    if (a.as_integer_class() >= p2 || a.as_integer_class() < 0)
        mp_fdiv_r(a_final, a.as_integer_class(), p2);
    if (a_final < 2)
        return true;
 
    if (!probab_prime_p(*integer(p2))) {
        if ((p2 % 2 == 1) && jacobi(*integer(a_final), p) == -1)
            return false;
 
        const RCP<const Integer> a1 = integer(a_final);
        const RCP<const Integer> p1 = integer(p2);
 
        map_integer_uint prime_mul;
        prime_factor_multiplicities(prime_mul, *p1);
        bool ret_val;
 
        for (const auto &it : prime_mul) {
            ret_val = _is_nthroot_mod_prime_power(
                a1->as_integer_class(), integer(2)->as_integer_class(),
                it.first->as_integer_class(), it.second);
            if (not ret_val)
                return false;
        }
        return true;
    }
 
    return mp_legendre(a_final, p2) == 1;
}
 
bool is_nth_residue(const Integer &a, const Integer &n, const Integer &mod)
/*
Returns true if ``a`` (mod ``mod``) is in the set of nth powers mod ``mod``,
i.e a % mod in set([i**n % mod for i in range(mod)]).
*/
{
    integer_class _mod = mod.as_integer_class();
 
    if (_mod == 0) {
        return false;
    } else if (_mod == 1) {
        return true;
    }
 
    if (_mod < 0)
        _mod = -(_mod);
 
    RCP<const Integer> mod2 = integer(_mod);
    map_integer_uint prime_mul;
    prime_factor_multiplicities(prime_mul, *mod2);
    bool ret_val;
 
    for (const auto &it : prime_mul) {
        ret_val = _is_nthroot_mod_prime_power(
            a.as_integer_class(), n.as_integer_class(),
            it.first->as_integer_class(), it.second);
        if (not ret_val)
            return false;
    }
    return true;
}
 
int mobius(const Integer &a)
{
    if (a.as_int() <= 0) {
        throw SymEngineException("mobius: Integer <= 0");
    }
    map_integer_uint prime_mul;
    bool is_square_free = true;
    prime_factor_multiplicities(prime_mul, a);
    auto num_prime_factors = prime_mul.size();
    for (const auto &it : prime_mul) {
        int p_freq = it.second;
        if (p_freq > 1) {
            is_square_free = false;
            break;
        }
    }
    if (!is_square_free) {
        return 0;
    } else if (num_prime_factors % 2 == 0) {
        return 1;
    } else {
        return -1;
    }
}
 
long mertens(const unsigned long a)
{
    long mertens = 0;
    for (unsigned long i = 1; i <= a; ++i) {
        mertens += mobius(*(integer(i)));
    }
    return mertens;
}
 
integer_class mp_polygonal_number(const integer_class &s,
                                  const integer_class &n)
{
    auto res = ((s - 2) * n * n - (s - 4) * n) / 2;
    return res;
}
 
integer_class mp_principal_polygonal_root(const integer_class &s,
                                          const integer_class &x)
{
    integer_class tmp;
    mp_pow_ui(tmp, s - 4, 2);
    integer_class root = mp_sqrt(8 * x * (s - 2) + tmp);
    integer_class n = (root + s - 4) / (2 * (s - 2));
    return n;
}
 
std::pair<integer_class, integer_class>
mp_perfect_power_decomposition(const integer_class &n, bool lowest_exponent)
{
    // From
    // https://codegolf.stackexchange.com/questions/1935/fastest-algorithm-for-decomposing-a-perfect-power
    unsigned long p = 2;
    integer_class intone, i, j, m, res;
    intone = 1;
    std::pair<integer_class, integer_class> respair;
    respair = std::make_pair(n, intone);
 
    while ((intone << p) <= n) {
        i = 2;
        j = n;
        while (j > i + 1) {
            m = (i + j) / 2;
            mp_pow_ui(res, m, p);
            if (res > n)
                j = m;
            else
                i = m;
        }
        mp_pow_ui(res, i, p);
        if (res == n) {
            respair = std::make_pair(i, p);
            if (lowest_exponent) {
                return respair;
            }
        }
        p++;
    }
    return respair;
}
 
} // namespace SymEngine