Teoria de Números - Tutorial Sage

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# Teoria de Números

O Sage possui extensa funcionalidade para teoria de números. Por exemplo, podemos fazer aritmética em \(\ZZ/N\ZZ\) da seguinte forma:

Sage

    sage: R = IntegerModRing(97)
    sage: a = R(2) / R(3)
    sage: a
    33
    sage: a.rational_reconstruction()
    2/3
    sage: b = R(47)
    sage: b^20052005
    50
    sage: b.modulus()
    97
    sage: b.is_square()
    True

Python

    >>> from sage.all import *
    >>> R = IntegerModRing(Integer(97))
    >>> a = R(Integer(2)) / R(Integer(3))
    >>> a
    33
    >>> a.rational_reconstruction()
    2/3
    >>> b = R(Integer(47))
    >>> b**Integer(20052005)
    50
    >>> b.modulus()
    97
    >>> b.is_square()
    True

O Sage contém funções comuns em teoria de números. Por exemplo,

Sage

    sage: gcd(515,2005)
    5
    sage: factor(2005)
    5 * 401
    sage: c = factorial(25); c
    15511210043330985984000000
    sage: [valuation(c,p) for p in prime_range(2,23)]
    [22, 10, 6, 3, 2, 1, 1, 1]
    sage: next_prime(2005)
    2011
    sage: previous_prime(2005)
    2003
    sage: divisors(28); sum(divisors(28)); 2*28
    [1, 2, 4, 7, 14, 28]
    56
    56

[Python]

    >>> from sage.all import *
    >>> gcd(Integer(515),Integer(2005))
    5
    >>> factor(Integer(2005))
    5 * 401
    >>> c = factorial(Integer(25)); c
    15511210043330985984000000
    >>> [valuation(c,p) for p in prime_range(Integer(2),Integer(23))]
    [22, 10, 6, 3, 2, 1, 1, 1]
    >>> next_prime(Integer(2005))
    2011
    >>> previous_prime(Integer(2005))
    2003
    >>> divisors(Integer(28)); sum(divisors(Integer(28))); Integer(2)*Integer(28)
    [1, 2, 4, 7, 14, 28]
    56
    56

Perfeito!

A função `sigma(n,k)` do Sage soma as \(k\)-ésimas potências dos divisores de `n`:

Sage

    sage: sigma(28,0); sigma(28,1); sigma(28,2)
    6
    56
    1050

Python

    >>> from sage.all import *
    >>> sigma(Integer(28),Integer(0)); sigma(Integer(28),Integer(1)); sigma(Integer(28),Integer(2))
    6
    56
    1050

A seguir ilustramos o algoritmo de Euclides estendido, a função \(\phi\) de Euler, e o teorema do resto Chinês:

Sage

    sage: d,u,v = xgcd(12,15)
    sage: d == u*12 + v*15
    True
    sage: n = 2005
    sage: inverse_mod(3,n)
    1337
    sage: 3 * 1337
    4011
    sage: prime_divisors(n)
    [5, 401]
    sage: phi = n*prod([1 - 1/p for p in prime_divisors(n)]); phi
    1600
    sage: euler_phi(n)
    1600
    sage: prime_to_m_part(n, 5)
    401

[Python]

    >>> from sage.all import *
    >>> d,u,v = xgcd(Integer(12),Integer(15))
    >>> d == u*Integer(12) + v*Integer(15)
    True
    >>> n = Integer(2005)
    >>> inverse_mod(Integer(3),n)
    1337
    >>> Integer(3) * Integer(1337)
    4011
    >>> prime_divisors(n)
    [5, 401]
    >>> phi = n*prod([Integer(1) - Integer(1)/p for p in prime_divisors(n)]); phi
    1600
    >>> euler_phi(n)
    1600
    >>> prime_to_m_part(n, Integer(5))
    401

Agora verificamos algo sobre o problema \(3n+1\).

Sage

    sage: n = 2005
    sage: for i in range(1000):
    ....:     n = 3*odd_part(n) + 1
    ....:     if odd_part(n)==1:
    ....:         print(i)
    ....:         break
    38

Python

    >>> from sage.all import *
    >>> n = Integer(2005)
    >>> for i in range(Integer(1000)):
    ...     n = Integer(3)*odd_part(n) + Integer(1)
    ...     if odd_part(n)==Integer(1):
    ...         print(i)
    ...         break
    38

Por fim, ilustramos o teorema do resto Chinês.

Sage

    sage: x = crt(2, 1, 3, 5); x
    11
    sage: x % 3  # x mod 3 = 2
    2
    sage: x % 5  # x mod 5 = 1
    1
    sage: [binomial(13,m) for m in range(14)]
    [1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1]
    sage: [binomial(13,m)%2 for m in range(14)]
    [1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1]
    sage: [kronecker(m,13) for m in range(1,13)]
    [1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1]
    sage: n = 10000; sum([moebius(m) for m in range(1,n)])
    -23
    sage: list(Partitions(4))
    [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]

[Python]

    >>> from sage.all import *
    >>> x = crt(Integer(2), Integer(1), Integer(3), Integer(5)); x
    11
    >>> x % Integer(3)  # x mod 3 = 2
    2
    >>> x % Integer(5)  # x mod 5 = 1
    1
    >>> [binomial(Integer(13),m) for m in range(Integer(14))]
    [1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1]
    >>> [binomial(Integer(13),m)%Integer(2) for m in range(Integer(14))]
    [1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1]
    >>> [kronecker(m,Integer(13)) for m in range(Integer(1),Integer(13))]
    [1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1]
    >>> n = Integer(10000); sum([moebius(m) for m in range(Integer(1),n)])
    -23
    >>> list(Partitions(Integer(4)))
    [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]

## Números \(p\)-ádicos

O corpo dos números \(p\)-ádicos está implementado em Sage. Note que uma vez que um corpo \(p\)-ádico é criado, você não pode alterar a sua precisão.

Sage

    sage: K = Qp(11); K
    11-adic Field with capped relative precision 20
    sage: a = K(211/17); a
    4 + 4*11 + 11^2 + 7*11^3 + 9*11^5 + 5*11^6 + 4*11^7 + 8*11^8 + 7*11^9
      + 9*11^10 + 3*11^11 + 10*11^12 + 11^13 + 5*11^14 + 6*11^15 + 2*11^16
      + 3*11^17 + 11^18 + 7*11^19 + O(11^20)
    sage: b = K(3211/11^2); b
    10*11^-2 + 5*11^-1 + 4 + 2*11 + O(11^18)

Python

    >>> from sage.all import *
    >>> K = Qp(Integer(11)); K
    11-adic Field with capped relative precision 20
    >>> a = K(Integer(211)/Integer(17)); a
    4 + 4*11 + 11^2 + 7*11^3 + 9*11^5 + 5*11^6 + 4*11^7 + 8*11^8 + 7*11^9
      + 9*11^10 + 3*11^11 + 10*11^12 + 11^13 + 5*11^14 + 6*11^15 + 2*11^16
      + 3*11^17 + 11^18 + 7*11^19 + O(11^20)
    >>> b = K(Integer(3211)/Integer(11)**Integer(2)); b
    10*11^-2 + 5*11^-1 + 4 + 2*11 + O(11^18)

Muito trabalho foi feito implementando anéis de inteiros em corpos \(p\)-ádicos ou corpos numéricos além de \(Z\). O leitor interessado é convidado a perguntar mais detalhes aos especialistas na lista `sage-support` no Google Groups.

Diversos métodos relacionados já estão implementados na classe NumberField.

Sage

    sage: R.<x> = PolynomialRing(QQ)
    sage: K = NumberField(x^3 + x^2 - 2*x + 8, 'a')
    sage: K.integral_basis()
    [1, 1/2*a^2 + 1/2*a, a^2]

[Python]

    >>> from sage.all import *
    >>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
    >>> K = NumberField(x**Integer(3) + x**Integer(2) - Integer(2)*x + Integer(8), 'a')
    >>> K.integral_basis()
    [1, 1/2*a^2 + 1/2*a, a^2]

Sage

    sage: K.galois_group()
    Galois group 3T2 (S3) with order 6 of x^3 + x^2 - 2*x + 8

Python

    >>> from sage.all import *
    >>> K.galois_group()
    Galois group 3T2 (S3) with order 6 of x^3 + x^2 - 2*x + 8

Sage

    sage: K.polynomial_quotient_ring()
    Univariate Quotient Polynomial Ring in a over Rational Field with modulus
    x^3 + x^2 - 2*x + 8
    sage: K.units()
    (-3*a^2 - 13*a - 13,)
    sage: K.discriminant()
    -503
    sage: K.class_group()
    Class group of order 1 of Number Field in a with
    defining polynomial x^3 + x^2 - 2*x + 8
    sage: K.class_number()
    1

[Python]

    >>> from sage.all import *
    >>> K.polynomial_quotient_ring()
    Univariate Quotient Polynomial Ring in a over Rational Field with modulus
    x^3 + x^2 - 2*x + 8
    >>> K.units()
    (-3*a^2 - 13*a - 13,)
    >>> K.discriminant()
    -503
    >>> K.class_group()
    Class group of order 1 of Number Field in a with
    defining polynomial x^3 + x^2 - 2*x + 8
    >>> K.class_number()
    1