gh-153222: Add math.integer.isprime() and math.integer.primes()

Draft implementation of prime-number functionality for the math.integer
module (deferred out of the initial PEP 791 scope), supporting integers
less than 2**64.

isprime() uses the deterministic Miller-Rabin test: bases {2, 7, 61}
below 4759123141, Jim Sinclair's seven bases up to 2**64 (verified
against the Feitsma-Galway list of base-2 strong pseudoprimes).  The
result is exact for the whole supported range; larger arguments raise
OverflowError.

primes(start=2, stop=None) returns an iterator of the primes in
[start, stop), unbounded if stop is None.

Neither function is mirrored into the math namespace (PEP 791).

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

Author: serhiy-storchaka

Doc/library/math.integer.rst

@@ -13,8 +13,9 @@ These functions accept integers and objects that implement the
 :meth:`~object.__index__` method which is used to convert the object to an integer
 number.
 
-The following functions are provided by this module.  All return values are
-computed exactly and are integers.
+The following functions are provided by this module.
+Unless stated otherwise below,
+all return values are computed exactly and are integers.
 
 
 .. function:: comb(n, k, /)
@@ -46,6 +47,20 @@ computed exactly and are integers.
    returns ``0``.
 
 
+.. function:: isprime(n, /)
+
+   Return ``True`` if *n* is a prime number, ``False`` otherwise.
+
+   A prime number is a natural number greater than 1
+   that is not a product of two smaller natural numbers.
+   Negative numbers, ``0`` and ``1`` are not prime.
+
+   The argument must be less than 2\ :sup:`64`;
+   raises :exc:`OverflowError` otherwise.
+
+   .. versionadded:: next
+
+
 .. function:: isqrt(n, /)
 
    Return the integer square root of the nonnegative integer *n*. This is the
@@ -83,3 +98,21 @@ computed exactly and are integers.
    and the function returns ``n!``.
 
    Raises :exc:`ValueError` if either of the arguments are negative.
+
+
+.. function:: primes(start=2, stop=None)
+
+   Return an iterator of the prime numbers *p* with ``start <= p < stop``,
+   in increasing order.
+   If *stop* is ``None`` (the default), the iteration does not stop.
+
+   Roughly equivalent to
+   ``(p for p in itertools.count(start) if isprime(p))``
+   for the unbounded form.
+
+   The bounds must be less than 2\ :sup:`64`;
+   raises :exc:`OverflowError` otherwise.
+   Unbounded iteration raises :exc:`OverflowError`
+   if the candidates reach that limit.
+
+   .. versionadded:: next

Doc/whatsnew/3.16.rst

@@ -279,6 +279,15 @@ math
   (Contributed by Jeff Epler in :gh:`150534`.)
 
 
+math.integer
+------------
+
+* Added :func:`math.integer.isprime` for primality testing of integers
+  less than 2**64 and
+  :func:`math.integer.primes` for iterating over prime numbers.
+  (Contributed by Serhiy Storchaka in :gh:`153222`.)
+
+
 os
 --
 

Include/internal/pycore_global_objects_fini_generated.h

@@ -820,6 +820,7 @@ struct _Py_global_strings {
         STRUCT_FOR_ID(stdout)
         STRUCT_FOR_ID(step)
         STRUCT_FOR_ID(steps)
+        STRUCT_FOR_ID(stop)
         STRUCT_FOR_ID(store_name)
         STRUCT_FOR_ID(strategy)
         STRUCT_FOR_ID(strftime)