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); a
1234
sage: b = Z(5678); b
5678
sage: type(a)
<class '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: 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()
Join of Category of euclidean domains
     and Category of infinite enumerated sets
     and Category of metric spaces
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 0o are interpreted as octal:

sage: parent('37')
<... 'str'>
sage: parent(Z('37'))
Integer Ring
sage: Z('0x10')
16
sage: Z('0x1a')
26
sage: Z('0o20')
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); a
1234
sage: b = Z(5678); b
5678
sage: type(a)
<class '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)
<class 'sage.rings.rational.Rational'>
sage: a/a
1
sage: type(a/a)
<class 'sage.rings.rational.Rational'>

For floor division, use the // operator instead:

sage: a // b
0
sage: type(a//b)
<class '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.

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 metric 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)
Integer Ring
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, **kwds)#

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())
<class '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])
<class 'sage.rings.integer.Integer'>

is_field(proof=True)#

Return False since the integers are not a field.

EXAMPLES:

sage: ZZ.is_field()
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

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, **kwds)#

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'
- 'gaussian'

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 \(2^x - 1\).

If the distribution 'gaussian' is specified, the output is sampled from a discrete Gaussian distribution with parameter \(\sigma=x\) and centered at zero. That is, the integer \(v\) is returned with probability proportional to \(\mathrm{exp}(-v^2/(2\sigma^2))\). See sage.stats.distributions.discrete_gaussian_integer for details. Note that if many samples from the same discrete Gaussian distribution are needed, it is faster to construct a sage.stats.distributions.discrete_gaussian_integer.DiscreteGaussianDistributionIntegerSampler object which is then repeatedly queried.

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().parent() is ZZ
True

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

sage: from collections import defaultdict
sage: def add_samples(*args, **kwds):
....:     global dic, counter
....:     for _ in range(100):
....:         counter += 1
....:         dic[ZZ.random_element(*args, **kwds)] += 1

sage: prob = lambda x : 1/5
sage: dic = defaultdict(Integer)
sage: counter = 0.0
sage: add_samples(distribution="uniform")
sage: while any(abs(dic[i]/counter - prob(i)) > 0.01 for i in dic):
....:     add_samples(distribution="uniform")

Here we use the distribution '1/n':

sage: def prob(n):
....:     if n == 0:
....:         return 1/5
....:     return 2/(5*abs(n)*(abs(n) + 1))
sage: dic = defaultdict(Integer)
sage: counter = 0.0
sage: add_samples(distribution="1/n")
sage: while any(abs(dic[i]/counter - prob(i)) > 0.01 for i in dic):
....:     add_samples(distribution="1/n")

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

sage: -10 <= ZZ.random_element(-10, 10) < 10
True
sage: prob = lambda x : 1/20
sage: dic = defaultdict(Integer)
sage: counter = 0.0
sage: add_samples(-10, 10)
sage: while any(abs(dic[i]/counter - prob(i)) > 0.01 for i in dic):
....:     add_samples(-10, 10)

sage: 0 <= ZZ.random_element(5) < 5
True
sage: prob = lambda x : 1/5
sage: dic = defaultdict(Integer)
sage: counter = 0.0
sage: add_samples(5)
sage: while any(abs(dic[i]/counter - prob(i)) > 0.01 for i in dic):
....:     add_samples(5)

sage: while ZZ.random_element(10^50) < 10^49:
....:     pass

Notice that the right endpoint is not included:

sage: all(ZZ.random_element(-2, 2) < 2 for _ in range(100))
True

We return a sample from a discrete Gaussian distribution:

sage: ZZ.random_element(11.0, distribution="gaussian").parent() is ZZ
True

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, names=None)#

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
names - ignored (for compatibility with number fields)

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

valuation(p)#

Return the discrete valuation with uniformizer p.

EXAMPLES:

sage: v = ZZ.valuation(3); v
3-adic valuation
sage: v(3)
1