"""
The set of prime numbers
AUTHORS:
- William Stein (2005): original version
- Florent Hivert (2009-11): adapted to the category framework. The following
methods were removed:
- cardinality, __len__, __iter__: provided by EnumeratedSets
- __cmp__(self, other): __eq__ is provided by UniqueRepresentation
and seems to do as good a job (all test pass)
"""
from sage.rings.all import ZZ
from set import Set_generic
from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets
from sage.rings.arith import nth_prime
from sage.structure.unique_representation import UniqueRepresentation
class Primes(Set_generic, UniqueRepresentation):
"""
The set of prime numbers.
EXAMPLES::
sage: P = Primes(); P
Set of all prime numbers: 2, 3, 5, 7, ...
We show various operations on the set of prime numbers::
sage: P.cardinality()
+Infinity
sage: R = Primes()
sage: P == R
True
sage: 5 in P
True
sage: 100 in P
False
sage: len(P) # note: this used to be a TypeError
Traceback (most recent call last):
...
NotImplementedError: infinite list
"""
@staticmethod
def __classcall__(cls, proof=True):
"""
TESTS::
sage: Primes(proof=True) is Primes()
True
sage: Primes(proof=False) is Primes()
False
"""
return super(Primes, cls).__classcall__(cls, proof)
def __init__(self, proof):
"""
EXAMPLES::
sage: P = Primes(); P
Set of all prime numbers: 2, 3, 5, 7, ...
sage: Q = Primes(proof = False); Q
Set of all prime numbers: 2, 3, 5, 7, ...
TESTS::
sage: P.category()
Category of facade infinite enumerated sets
sage: TestSuite(P).run()
sage: Q.category()
Category of facade infinite enumerated sets
sage: TestSuite(Q).run()
The set of primes can be compared to various things,
but is only equal to itself::
sage: P = Primes()
sage: R = Primes()
sage: cmp(P,R)
0
sage: P == R
True
sage: P != R
False
sage: Q=[1,2,3]
sage: Q != P # indirect doctest
True
sage: R.<x>=ZZ[]
sage: P!=x^2+x
True
Make sure changing order changes the comparison with something
of a different type::
sage: cmp('foo', Primes()) != cmp(Primes(), 'foo')
True
"""
super(Primes, self).__init__(facade = ZZ, category = InfiniteEnumeratedSets())
self.__proof = proof
def _repr_(self):
"""
Representation of the set of primes.
EXAMPLES::
sage: P = Primes(); P
Set of all prime numbers: 2, 3, 5, 7, ...
"""
return "Set of all prime numbers: 2, 3, 5, 7, ..."
def __contains__(self, x):
"""
Checks whether an object is a prime number.
EXAMPLES::
sage: P = Primes()
sage: 5 in P
True
sage: 100 in P
False
sage: 1.5 in P
False
sage: e in P
False
"""
try:
if not x in ZZ:
return False
return ZZ(x).is_prime()
except TypeError:
return False
def _an_element_(self):
"""
Returns a typical prime number.
EXAMPLES::
sage: P = Primes()
sage: P._an_element_()
43
"""
return ZZ(43)
def first(self):
"""
Returns the first prime number.
EXAMPLES::
sage: P = Primes()
sage: P.first()
2
"""
return ZZ(2)
def next(self, pr):
"""
Returns the next prime number.
EXAMPLES::
sage: P = Primes()
sage: P.next(5)
7
"""
pr = pr.next_prime(self.__proof)
return pr
def unrank(self, n):
"""
Returns the n-th prime number.
EXAMPLES::
sage: P = Primes()
sage: P.unrank(0)
2
sage: P.unrank(5)
13
sage: P.unrank(42)
191
"""
return nth_prime(n+1)
