Ring \(\ZZ\) of Integers

Ring \(\ZZ\) of Integers

The IntegerRing_class represents the ring \(\ZZ\) of (arbitrary precision) integers. Each integer is an instance of Integer, which is defined in a Pyrex extension module that wraps GMP integers (the mpz_t type in GMP).

sage: Z = IntegerRing(); Z
Integer Ring
sage: Z.characteristic()
0
sage: Z.is_field()
False

There is a unique instance of the integer ring. To create an Integer, coerce either a Python int, long, or a string. Various other types will also coerce to the integers, when it makes sense.

sage: a = Z(1234); b = Z(5678); print a, b
1234 5678
sage: type(a)
<type 'sage.rings.integer.Integer'>
sage: a + b
6912
sage: Z('94803849083985934859834583945394')
94803849083985934859834583945394
sage.rings.integer_ring.IntegerRing()

Return the integer ring.

EXAMPLES:

sage: IntegerRing()
Integer Ring
sage: ZZ==IntegerRing()
True
class sage.rings.integer_ring.IntegerRing_class

Bases: sage.rings.ring.PrincipalIdealDomain

The ring of integers.

In order to introduce the ring \(\ZZ\) of integers, we illustrate creation, calling a few functions, and working with its elements.

sage: Z = IntegerRing(); Z
Integer Ring
sage: Z.characteristic()
0
sage: Z.is_field()
False
sage: Z.category()
Category of euclidean domains
sage: Z(2^(2^5) + 1)
4294967297

One can give strings to create integers. Strings starting with 0x are interpreted as hexadecimal, and strings starting with 0 are interpreted as octal:

sage: parent('37')
<type 'str'>
sage: parent(Z('37'))
Integer Ring
sage: Z('0x10')
16
sage: Z('0x1a')
26
sage: Z('020')
16

As an inverse to digits(), lists of digits are accepted, provided that you give a base. The lists are interpreted in little-endian order, so that entry i of the list is the coefficient of base^i:

sage: Z([4,1,7],base=100)
70104
sage: Z([4,1,7],base=10)
714
sage: Z([3, 7], 10)
73
sage: Z([3, 7], 9)
66
sage: Z([], 10)
0

Alphanumeric strings can be used for bases 2..36; letters a to z represent numbers 10 to 36. Letter case does not matter.

sage: Z("sage",base=32)
928270
sage: Z("SAGE",base=32)
928270
sage: Z("Sage",base=32)
928270
sage: Z([14, 16, 10, 28],base=32)
928270
sage: 14 + 16*32 + 10*32^2 + 28*32^3
928270

We next illustrate basic arithmetic in \(\ZZ\):

sage: a = Z(1234); b = Z(5678); print a, b
1234 5678
sage: type(a)
<type 'sage.rings.integer.Integer'>
sage: a + b
6912
sage: b + a
6912
sage: a * b
7006652
sage: b * a
7006652
sage: a - b
-4444
sage: b - a
4444

When we divide two integers using /, the result is automatically coerced to the field of rational numbers, even if the result is an integer.

sage: a / b
617/2839
sage: type(a/b)
<type 'sage.rings.rational.Rational'>
sage: a/a
1
sage: type(a/a)
<type 'sage.rings.rational.Rational'>

For floor division, use the // operator instead:

sage: a // b
0
sage: type(a//b)
<type 'sage.rings.integer.Integer'>

Next we illustrate arithmetic with automatic coercion. The types that coerce are: str, int, long, Integer.

sage: a + 17
1251
sage: a * 374
461516
sage: 374 * a
461516
sage: a/19
1234/19
sage: 0 + Z(-64)
-64

Integers can be coerced:

sage: a = Z(-64)
sage: int(a)
-64

We can create integers from several types of objects:

sage: Z(17/1)
17
sage: Z(Mod(19,23))
19
sage: Z(2 + 3*5 + O(5^3))
17

Arbitrary numeric bases are supported; strings or list of integers are used to provide the digits (more details in IntegerRing_class):

sage: Z("sage",base=32)
928270
sage: Z([14, 16, 10, 28],base=32)
928270

The digits method allows you to get the list of digits of an integer in a different basis (note that the digits are returned in little-endian order):

sage: b = Z([4,1,7],base=100)
sage: b
70104
sage: b.digits(base=71)
[27, 64, 13]

sage: Z(15).digits(2)
[1, 1, 1, 1]
sage: Z(15).digits(3)
[0, 2, 1]

The str method returns a string of the digits, using letters a to z to represent digits 10..36:

sage: Z(928270).str(base=32)
'sage'

Note that str only works with bases 2 through 36.

TESTS:

sage: TestSuite(ZZ).run()
absolute_degree()

Return the absolute degree of the integers, which is 1.

Here, absolute degree refers to the rank of the ring as a module over the integers.

EXAMPLES:

sage: ZZ.absolute_degree()
1
characteristic()

Return the characteristic of the integers, which is 0.

EXAMPLES:

sage: ZZ.characteristic()
0
completion(p, prec, extras={})

Return the completion of the integers at the prime \(p\).

INPUT:

  • p – a prime (or infinity)
  • prec – the desired precision
  • extras – any further parameters to pass to the method used to create the completion.

OUTPUT:

  • The completion of \(\ZZ\) at \(p\).

EXAMPLES:

sage: ZZ.completion(infinity, 53)
Real Field with 53 bits of precision
sage: ZZ.completion(5, 15, {'print_mode': 'bars'})
5-adic Ring with capped relative precision 15
degree()

Return the degree of the integers, which is 1.

Here, degree refers to the rank of the ring as a module over the integers.

EXAMPLES:

sage: ZZ.degree()
1
extension(poly, names=None, embedding=None)

Return the order generated by the specified list of polynomials.

INPUT:

  • poly – a list of one or more polynomials
  • names – a parameter which will be passed to EquationOrder().
  • embedding – a parameter which will be passed to EquationOrder().

OUTPUT:

  • Given a single polynomial as input, return the order generated by a root of the polynomial in the field generated by a root of the polynomial.

    Given a list of polynomials as input, return the relative order generated by a root of the first polynomial in the list, over the order generated by the roots of the subsequent polynomials.

EXAMPLES:

sage: ZZ.extension(x^2-5, 'a')
Order in Number Field in a with defining polynomial x^2 - 5
sage: ZZ.extension([x^2 + 1, x^2 + 2], 'a,b')
Relative Order in Number Field in a with defining polynomial
x^2 + 1 over its base field
fraction_field()

Return the field of rational numbers - the fraction field of the integers.

EXAMPLES:

sage: ZZ.fraction_field()
Rational Field
sage: ZZ.fraction_field() == QQ
True
gen(n=0)

Return the additive generator of the integers, which is 1.

INPUT:

  • n (default: 0) – In a ring with more than one generator, the optional parameter \(n\) indicates which generator to return; since there is only one generator in this case, the only valid value for \(n\) is 0.

EXAMPLES:

sage: ZZ.gen()
1
sage: type(ZZ.gen())
<type 'sage.rings.integer.Integer'>
gens()

Return the tuple (1,) containing a single element, the additive generator of the integers, which is 1.

EXAMPLES:

sage: ZZ.gens(); ZZ.gens()[0]
(1,)
1
sage: type(ZZ.gens()[0])
<type 'sage.rings.integer.Integer'>
is_field(proof=True)

Return False since the integers are not a field.

EXAMPLES:

sage: ZZ.is_field()
False
is_finite()

Return False since the integers are an infinite ring.

EXAMPLES:

sage: ZZ.is_finite()
False
is_integrally_closed()

Return that the integer ring is, in fact, integrally closed.

EXAMPLES:

sage: ZZ.is_integrally_closed()
True
is_noetherian()

Return True since the integers are a Noetherian ring.

EXAMPLES:

sage: ZZ.is_noetherian()
True
is_subring(other)

Return True if \(\ZZ\) is a subring of other in a natural way.

Every ring of characteristic \(0\) contains \(\ZZ\) as a subring.

EXAMPLES:

sage: ZZ.is_subring(QQ)
True
krull_dimension()

Return the Krull dimension of the integers, which is 1.

EXAMPLES:

sage: ZZ.krull_dimension()
1
ngens()

Return the number of additive generators of the ring, which is 1.

EXAMPLES:

sage: ZZ.ngens()
1
sage: len(ZZ.gens())
1
order()

Return the order (cardinality) of the integers, which is +Infinity.

EXAMPLES:

sage: ZZ.order()
+Infinity
parameter()

Return an integer of degree 1 for the Euclidean property of \(\ZZ\), namely 1.

EXAMPLES:

sage: ZZ.parameter()
1
quotient(I, names=None)

Return the quotient of \(\ZZ\) by the ideal or integer I.

EXAMPLES:

sage: ZZ.quo(6*ZZ)
Ring of integers modulo 6
sage: ZZ.quo(0*ZZ)
Integer Ring
sage: ZZ.quo(3)
Ring of integers modulo 3
sage: ZZ.quo(3*QQ)
Traceback (most recent call last):
...
TypeError: I must be an ideal of ZZ
random_element(x=None, y=None, distribution=None)

Return a random integer.

INPUT:

  • x, y integers – bounds for the result.

  • distribution– a string:
    • 'uniform'
    • 'mpz_rrandomb'
    • '1/n'

OUTPUT:

  • With no input, return a random integer.

    If only one integer \(x\) is given, return an integer between 0 and \(x-1\).

    If two integers are given, return an integer between \(x\) and \(y-1\) inclusive.

    If at least one bound is given, the default distribution is the uniform distribution; otherwise, it is the distribution described below.

    If the distribution '1/n' is specified, the bounds are ignored.

    If the distribution 'mpz_rrandomb' is specified, the output is in the range from 0 to \(2^x - 1\).

The default distribution for ZZ.random_element() is based on \(X = \mbox{trunc}(4/(5R))\), where \(R\) is a random variable uniformly distributed between \(-1\) and \(1\). This gives \(\mbox{Pr}(X = 0) = 1/5\), and \(\mbox{Pr}(X = n) = 2/(5|n|(|n|+1))\) for \(n \neq 0\). Most of the samples will be small; \(-1\), \(0\), and \(1\) occur with probability \(1/5\) each. But we also have a small but non-negligible proportion of “outliers”; \(\mbox{Pr}(|X| \geq n) = 4/(5n)\), so for instance, we expect that \(|X| \geq 1000\) on one in 1250 samples.

We actually use an easy-to-compute truncation of the above distribution; the probabilities given above hold fairly well up to about \(|n| = 10000\), but around \(|n| = 30000\) some values will never be returned at all, and we will never return anything greater than \(2^{30}\).

EXAMPLES:

sage: [ZZ.random_element() for _ in range(10)]
[-8, 2, 0, 0, 1, -1, 2, 1, -95, -1]

The default uniform distribution is integers in \([-2, 2]\):

sage: [ZZ.random_element(distribution="uniform") for _ in range(10)]
[2, -2, 2, -2, -1, 1, -1, 2, 1, 0]

Here we use the distribution '1/n':

sage: [ZZ.random_element(distribution="1/n") for _ in range(10)]
[-6, 1, -1, 1, 1, -1, 1, -1, -3, 1]

If a range is given, the default distribution is uniform in that range:

sage: ZZ.random_element(-10,10)
-2
sage: ZZ.random_element(10)
2
sage: ZZ.random_element(10^50)
9531604786291536727294723328622110901973365898988
sage: [ZZ.random_element(5) for _ in range(10)]
[3, 1, 2, 3, 0, 0, 3, 4, 0, 3]

Notice that the right endpoint is not included:

sage: [ZZ.random_element(-2,2) for _ in range(10)]
[1, -2, -2, -1, -2, -1, -1, -2, 0, -2]

We compute a histogram over 1000 samples of the default distribution:

sage: from collections import defaultdict
sage: d = defaultdict(lambda: 0)
sage: for _ in range(1000):
...       samp = ZZ.random_element()
...       d[samp] = d[samp] + 1

sage: sorted(d.items())
[(-1955, 1), (-1026, 1), (-357, 1), (-248, 1), (-145, 1), (-81, 1), (-80, 1), (-79, 1), (-75, 1), (-69, 1), (-68, 1), (-63, 2), (-61, 1), (-57, 1), (-50, 1), (-37, 1), (-35, 1), (-33, 1), (-29, 2), (-27, 1), (-25, 1), (-23, 2), (-22, 3), (-20, 1), (-19, 1), (-18, 1), (-16, 4), (-15, 3), (-14, 1), (-13, 2), (-12, 2), (-11, 2), (-10, 7), (-9, 3), (-8, 3), (-7, 7), (-6, 8), (-5, 13), (-4, 24), (-3, 34), (-2, 75), (-1, 206), (0, 208), (1, 189), (2, 63), (3, 35), (4, 13), (5, 11), (6, 10), (7, 4), (8, 3), (10, 1), (11, 1), (12, 1), (13, 1), (14, 1), (16, 3), (18, 2), (19, 1), (26, 2), (27, 1), (28, 2), (29, 1), (30, 1), (32, 1), (33, 2), (35, 1), (37, 1), (39, 1), (41, 1), (42, 1), (52, 1), (91, 1), (94, 1), (106, 1), (111, 1), (113, 2), (132, 1), (134, 1), (232, 1), (240, 1), (2133, 1), (3636, 1)]
range(start, end=None, step=None)

Optimized range function for Sage integers.

AUTHORS:

  • Robert Bradshaw (2007-09-20)

EXAMPLES:

sage: ZZ.range(10)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
sage: ZZ.range(-5,5)
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4]
sage: ZZ.range(0,50,5)
[0, 5, 10, 15, 20, 25, 30, 35, 40, 45]
sage: ZZ.range(0,50,-5)
[]
sage: ZZ.range(50,0,-5)
[50, 45, 40, 35, 30, 25, 20, 15, 10, 5]
sage: ZZ.range(50,0,5)
[]
sage: ZZ.range(50,-1,-5)
[50, 45, 40, 35, 30, 25, 20, 15, 10, 5, 0]

It uses different code if the step doesn’t fit in a long:

sage: ZZ.range(0,2^83,2^80)
[0, 1208925819614629174706176, 2417851639229258349412352, 3626777458843887524118528, 4835703278458516698824704, 6044629098073145873530880, 7253554917687775048237056, 8462480737302404222943232]

Make sure trac ticket #8818 is fixed:

sage: ZZ.range(1r, 10r)
[1, 2, 3, 4, 5, 6, 7, 8, 9]
residue_field(prime, check=True)

Return the residue field of the integers modulo the given prime, i.e. \(\ZZ/p\ZZ\).

INPUT:

  • prime - a prime number
  • check - (boolean, default True) whether or not to check the primality of prime.

OUTPUT: The residue field at this prime.

EXAMPLES:

sage: F = ZZ.residue_field(61); F
Residue field of Integers modulo 61
sage: pi = F.reduction_map(); pi
Partially defined reduction map:
  From: Rational Field
  To:   Residue field of Integers modulo 61
sage: pi(123/234)
6
sage: pi(1/61)
Traceback (most recent call last):
...
ZeroDivisionError: Cannot reduce rational 1/61 modulo 61:
it has negative valuation
sage: lift = F.lift_map(); lift
Lifting map:
  From: Residue field of Integers modulo 61
  To:   Integer Ring
sage: lift(F(12345/67890))
33
sage: (12345/67890) % 61
33

Construction can be from a prime ideal instead of a prime:

sage: ZZ.residue_field(ZZ.ideal(97))
Residue field of Integers modulo 97

TESTS:

sage: ZZ.residue_field(ZZ.ideal(96))
Traceback (most recent call last):
...
TypeError: Principal ideal (96) of Integer Ring is not prime
sage: ZZ.residue_field(96)
Traceback (most recent call last):
...
TypeError: 96 is not prime
zeta(n=2)

Return a primitive n-th root of unity in the integers, or raise an error if none exists.

INPUT:

  • n – (default 2) a positive integer

OUTPUT:

  • an n-th root of unity in \(\ZZ\).

EXAMPLE:

sage: ZZ.zeta()
-1
sage: ZZ.zeta(1)
1
sage: ZZ.zeta(3)
Traceback (most recent call last):
...
ValueError: no nth root of unity in integer ring
sage: ZZ.zeta(0)
Traceback (most recent call last):
...
ValueError: n must be positive in zeta()
sage.rings.integer_ring.crt_basis(X, xgcd=None)

Compute and return a Chinese Remainder Theorem basis for the list X of coprime integers.

INPUT:

  • X – a list of Integers that are coprime in pairs.
  • xgcd – an optional parameter which is ignored.

OUTPUT:

  • E - a list of Integers such that E[i] = 1 (mod X[i]) and E[i] = 0 (mod X[j]) for all \(j \neq i\).

For this explanation, let E[i] be denoted by \(E_i\).

The \(E_i\) have the property that if \(A\) is a list of objects, e.g., integers, vectors, matrices, etc., where \(A_i\) is understood modulo \(X_i\), then a CRT lift of \(A\) is simply

\[\sum_i E_i A_i.\]

ALGORITHM: To compute \(E_i\), compute integers \(s\) and \(t\) such that

\[s X_i + t \prod_{i \neq j} X_j = 1. (\*)\]

Then

\[E_i = t \prod_{i \neq j} X[j].\]

Notice that equation (*) implies that \(E_i\) is congruent to 1 modulo \(X_i\) and to 0 modulo the other \(X_j\) for \(j \neq i\).

COMPLEXITY: We compute len(X) extended GCD’s.

EXAMPLES:

sage: X = [11,20,31,51]
sage: E = crt_basis([11,20,31,51])
sage: E[0]%X[0], E[1]%X[0], E[2]%X[0], E[3]%X[0]
(1, 0, 0, 0)
sage: E[0]%X[1], E[1]%X[1], E[2]%X[1], E[3]%X[1]
(0, 1, 0, 0)
sage: E[0]%X[2], E[1]%X[2], E[2]%X[2], E[3]%X[2]
(0, 0, 1, 0)
sage: E[0]%X[3], E[1]%X[3], E[2]%X[3], E[3]%X[3]
(0, 0, 0, 1)
sage.rings.integer_ring.is_IntegerRing(x)

Internal function: return True iff x is the ring \(\ZZ\) of integers.

TESTS:

sage: from sage.rings.integer_ring import is_IntegerRing
sage: is_IntegerRing(ZZ)
True
sage: is_IntegerRing(QQ)
False
sage: is_IntegerRing(parent(3))
True
sage: is_IntegerRing(parent(1/3))
False

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Elements of the ring \(\ZZ\) of integers

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