Coverage for local/lib/python2.7/site-packages/sage/sets/primes.py : 42%

""" 

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 __future__ import absolute_import 

#***************************************************************************** 

# Copyright (C) 2005 William Stein <wstein@gmail.com> 

# 2009 Florent Hivert <Florent.Hivert@univ-rouen.fr> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from sage.rings.all import ZZ 

from .set import Set_generic 

from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets 

from sage.arith.all 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) 

Traceback (most recent call last): 

... 

NotImplementedError: infinite set 

""" 

@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: 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 

""" 

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): 

""" 

Check 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 x not in ZZ: 

return False 

return ZZ(x).is_prime() 

except TypeError: 

return False 

def _an_element_(self): 

""" 

Return a typical prime number. 

EXAMPLES:: 

sage: P = Primes() 

sage: P._an_element_() 

43 

""" 

return ZZ(43) 

def first(self): 

""" 

Return the first prime number. 

EXAMPLES:: 

sage: P = Primes() 

sage: P.first() 

2 

""" 

return ZZ(2) 

def next(self, pr): 

""" 

Return 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): 

""" 

Return 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)