# 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)#

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
```
first()#

Return the first prime number.

EXAMPLES:

```sage: P = Primes()
sage: P.first()
2
```
next(pr)#

Return the next prime number.

EXAMPLES:

```sage: P = Primes()
sage: P.next(5)                                                             # needs sage.libs.pari
7
```
unrank(n)#

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
```