The set of prime numbers¶
AUTHORS:
William Stein (2005): original version
Florent Hivert (2009-11): adapted to the category framework.
- class sage.sets.primes.Primes(proof)[source]¶
Bases:
Set_generic,UniqueRepresentationThe set of prime numbers.
EXAMPLES:
sage: P = Primes(); P Set of all prime numbers: 2, 3, 5, 7, ...
>>> from sage.all import * >>> P = Primes(); P Set of all prime numbers: 2, 3, 5, 7, ...
P = Primes(); P
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
>>> from sage.all import * >>> P.cardinality() +Infinity >>> R = Primes() >>> P == R True >>> Integer(5) in P True >>> Integer(100) in P False >>> len(P) Traceback (most recent call last): ... NotImplementedError: infinite set
P.cardinality() R = Primes() P == R 5 in P 100 in P len(P)
- first()[source]¶
Return the first prime number.
EXAMPLES:
sage: P = Primes() sage: P.first() 2
>>> from sage.all import * >>> P = Primes() >>> P.first() 2
P = Primes() P.first()
- next(pr)[source]¶
Return the next prime number.
EXAMPLES:
sage: P = Primes() sage: P.next(5) # needs sage.libs.pari 7
>>> from sage.all import * >>> P = Primes() >>> P.next(Integer(5)) # needs sage.libs.pari 7
P = Primes() P.next(5) # needs sage.libs.pari
- unrank(n)[source]¶
Return the n-th prime number.
EXAMPLES:
sage: P = Primes() sage: P.unrank(0) # needs sage.libs.pari 2 sage: P.unrank(5) # needs sage.libs.pari 13 sage: P.unrank(42) # needs sage.libs.pari 191
>>> from sage.all import * >>> P = Primes() >>> P.unrank(Integer(0)) # needs sage.libs.pari 2 >>> P.unrank(Integer(5)) # needs sage.libs.pari 13 >>> P.unrank(Integer(42)) # needs sage.libs.pari 191
P = Primes() P.unrank(0) # needs sage.libs.pari P.unrank(5) # needs sage.libs.pari P.unrank(42) # needs sage.libs.pari