# Non Negative Integers¶

class sage.sets.non_negative_integers.NonNegativeIntegers(category=None)

The enumerated set of non negative integers.

This class implements the set of non negative integers, as an enumerated set (see InfiniteEnumeratedSets).

EXAMPLES:

sage: NN = NonNegativeIntegers()
sage: NN
Non negative integers
sage: NN.cardinality()
+Infinity
sage: TestSuite(NN).run()
sage: NN.list()
Traceback (most recent call last):
...
NotImplementedError: infinite list
sage: NN.element_class
<type 'sage.rings.integer.Integer'>
sage: it = iter(NN)
sage: [next(it), next(it), next(it), next(it), next(it)]
[0, 1, 2, 3, 4]
sage: NN.first()
0


Currently, this is just a “facade” parent; namely its elements are plain Sage Integers with Integer Ring as parent:

sage: x = NN(15); type(x)
<type 'sage.rings.integer.Integer'>
sage: x.parent()
Integer Ring
sage: x+3
18


In a later version, there will be an option to specify whether the elements should have Integer Ring or Non negative integers as parent:

sage: NN = NonNegativeIntegers(facade = False) # todo: not implemented
sage: x = NN(5)                                # todo: not implemented
sage: x.parent()                               # todo: not implemented
Non negative integers


This runs generic sanity checks on NN:

sage: TestSuite(NN).run()


TODO: do not use NN any more in the doctests for NonNegativeIntegers.

Element

alias of Integer

an_element()

EXAMPLES:

sage: NonNegativeIntegers().an_element()
42

from_integer

alias of Integer

next(o)

EXAMPLES:

sage: NN = NonNegativeIntegers()
sage: NN.next(3)
4

some_elements()

EXAMPLES:

sage: NonNegativeIntegers().some_elements()
[0, 1, 3, 42]

unrank(rnk)

EXAMPLES:

sage: NN = NonNegativeIntegers()
sage: NN.unrank(100)
100


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Positive Integers

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The set of prime numbers