数論#
Sageは数論関連の多彩な機能を備えている． 例えば，以下のようにして \(\ZZ/N\ZZ\) 上の演算を実行することができる:
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
>>> 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
Sageは数論では標準となっている関数群を装備している．例えば
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
>>> 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
という具合で，申し分なしだ．
Sageの sigma(n,k) 関数は， n の商の \(k\) 乗の和を計算する:
sage: sigma(28,0); sigma(28,1); sigma(28,2)
6
56
1050
>>> from sage.all import *
>>> sigma(Integer(28),Integer(0)); sigma(Integer(28),Integer(1)); sigma(Integer(28),Integer(2))
6
56
1050
以下では，拡張ユークリッド互除法，オイラーの \(\phi\) -関数，そして中国剰余定理を見てみよう:
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
>>> 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
次に， \(3n+1\) 問題をちょっと調べてみる．
sage: n = 2005
sage: for i in range(1000):
....: n = 3 * odd_part(n) + 1
....: if odd_part(n) == 1:
....: print(i)
....: break
38
>>> 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
最後に，中国剰余定理を確かめてみよう．
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: Partitions(4).list()
[[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]
>>> 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
>>> Partitions(Integer(4)).list()
[[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]
\(p\) -進数#
Sageには \(p\) -進数体も組込まれている． ただし，いったん生成された \(p\) -進数体については，後でその精度を変更することはできないことを注意しておく．
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)
>>> 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)
\(p\) -進数体あるいは \(QQ\) 以外の数体上に整数環を実装するために多大の労力が投入されてきている．
興味ある読者はGoogleグループ sage-support で専門家に詳細を聞いてみてほしい．
NumberField クラスには，すでに多くの関連メソッドが実装されている．
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]
>>> 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: K.galois_group()
Galois group 3T2 (S3) with order 6 of x^3 + x^2 - 2*x + 8
>>> from sage.all import *
>>> K.galois_group()
Galois group 3T2 (S3) with order 6 of x^3 + x^2 - 2*x + 8
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
>>> 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