Standard Commutative Rings

Integers

• Ring \(\ZZ\) of Integers
  • IntegerRing()
  • IntegerRing_class
  • crt_basis()
  • is_IntegerRing()
• Elements of the ring \(\ZZ\) of integers
  • GCD_list()
  • Integer
  • IntegerWrapper
  • free_integer_pool()
  • int_to_Z
  • is_Integer()
  • long_to_Z
  • make_integer()
• Cython wrapper for bernmm library
  • bernmm_bern_modp()
  • bernmm_bern_rat()
• Bernoulli numbers modulo p
  • bernoulli_mod_p()
  • bernoulli_mod_p_single()
  • verify_bernoulli_mod_p()
• Integer factorization functions
  • aurifeuillian()
  • factor_aurifeuillian()
  • factor_cunningham()
  • factor_trial_division()
  • factor_using_pari()
• Basic arithmetic with C integers
  • arith_int
  • arith_llong
  • prime_range()
• Fast decomposition of small integers into sums of squares
  • four_squares_pyx()
  • is_sum_of_two_squares_pyx()
  • three_squares_pyx()
  • two_squares_pyx()
• Fast Arithmetic Functions
  • LCM_list()
  • lcm()
• Generic implementation of powering
  • generic_power()
• Utility classes for multi-modular algorithms
  • MultiModularBasis
  • MultiModularBasis_base
  • MutableMultiModularBasis
• Miscellaneous arithmetic functions
  • CRT()
  • CRT_basis()
  • CRT_list()
  • CRT_vectors()
  • Euler_Phi
  • GCD()
  • Moebius
  • Sigma
  • XGCD()
  • algdep()
  • algebraic_dependency()
  • bernoulli()
  • binomial()
  • binomial_coefficients()
  • continuant()
  • crt()
  • dedekind_psi()
  • dedekind_sum()
  • differences()
  • divisors()
  • eratosthenes()
  • factor()
  • factorial()
  • falling_factorial()
  • four_squares()
  • fundamental_discriminant()
  • gauss_sum()
  • gcd()
  • get_gcd()
  • get_inverse_mod()
  • hilbert_conductor()
  • hilbert_conductor_inverse()
  • hilbert_symbol()
  • integer_ceil()
  • integer_floor()
  • integer_trunc()
  • inverse_mod()
  • is_power_of_two()
  • is_prime()
  • is_prime_power()
  • is_pseudoprime()
  • is_pseudoprime_power()
  • is_square()
  • is_squarefree()
  • jacobi_symbol()
  • kronecker()
  • kronecker_symbol()
  • legendre_symbol()
  • mqrr_rational_reconstruction()
  • multinomial()
  • multinomial_coefficients()
  • next_prime()
  • next_prime_power()
  • next_probable_prime()
  • nth_prime()
  • number_of_divisors()
  • odd_part()
  • power_mod()
  • previous_prime()
  • previous_prime_power()
  • prime_divisors()
  • prime_factors()
  • prime_powers()
  • prime_to_m_part()
  • primes()
  • primes_first_n()
  • primitive_root()
  • quadratic_residues()
  • radical()
  • random_prime()
  • rational_reconstruction()
  • rising_factorial()
  • sort_complex_numbers_for_display()
  • squarefree_divisors()
  • subfactorial()
  • sum_of_k_squares()
  • three_squares()
  • trial_division()
  • two_squares()
  • valuation()
  • xgcd()
  • xkcd()
  • xlcm()

See also

• sage.sets.integer_range

• sage.sets.positive_integers

• sage.sets.non_negative_integers

• sage.sets.primes

Rationals

• Field \(\QQ\) of Rational Numbers
  • Q
  • QQ
  • RationalField
  • frac()
  • is_RationalField()
• Rational Numbers
  • Q_to_Z
  • Rational
  • Z_to_Q
  • int_to_Q
  • integer_rational_power()
  • is_Rational()
  • long_to_Q
  • make_rational()
  • rational_power_parts()

Indices and Tables

• Index

• Module Index

• Search Page