Subsets of primes defined by congruence conditions #41122

xcaruso · 2025-11-01T10:39:55Z

We implement subsets of primes given by congruence conditions and operations on those sets.

Here is a short demo.

sage: P = Primes(modulus=4); P
Set of prime numbers congruent to 1 modulo 4: 5, 13, 17, 29, ...
sage: P[:10]
[5, 13, 17, 29, 37, 41, 53, 61, 73, 89]
sage: P.next(500)
509
sage: for p in P:
....:     if p > 50: break
....:     print(p)
5
13
17
29
37
41

sage: P = Primes(modulus=4); P
Set of prime numbers congruent to 1 modulo 4: 5, 13, 17, 29, ...
sage: Q = Primes(modulus=4, classes=[3])
Set of prime numbers congruent to 3 modulo 4: 3, 7, 11, 19, ...
sage: PQ = P.union(Q)
sage: PQ
Set of all prime numbers with 2 excluded: 3, 5, 7, 11, ...
sage: PQ == Primes().exclude(2)
True

Cc @FloFuer

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The title is concise and informative.
The description explains in detail what this PR is about.
I have linked a relevant issue or discussion.
I have created tests covering the changes.
I have updated the documentation and checked the documentation preview.

github-actions · 2025-11-01T11:52:07Z

Documentation preview for this PR (built with commit 1ef8180; changes) is ready! 🎉
This preview will update shortly after each push to this PR.

src/sage/sets/primes.py

Co-authored-by: Martin Rubey <[email protected]>

src/sage/sets/primes.py

Co-authored-by: Martin Rubey <[email protected]>

xcaruso · 2025-11-02T04:57:35Z

Question: do we want methods for accessing modulus, classes and exceptions (or maybe included, excluded)?

xcaruso · 2025-11-02T07:29:53Z

Question: do we want methods for accessing modulus, classes and exceptions (or maybe included, excluded)?

Ok, I did it :-)

src/sage/sets/primes.py

mantepse · 2025-11-02T17:19:02Z

src/sage/sets/primes.py

+        else:
+            return P == other


Suggested change

else:

return P == other

return P == other

I know that it is equivalent but I prefer keeping the else here because I find it more logic.

src/sage/sets/primes.py

Co-authored-by: Martin Rubey <[email protected]>

…mes_congruences

src/sage/sets/primes.py

Co-authored-by: Martin Rubey <[email protected]>

mantepse · 2025-11-02T23:06:03Z

I have one more comment: I am quite sure it would be better to get rid of in_range and make intersection and also union more general, by allowing other to be a (finite) iterable. Besides, in_range currently fails to deal with finite sets of primes.

Thus, the most naive version would be as below. I am sure one can do better - in the worst case make a special case for range objects.

index 6eb5d2a52f6..e4c1480cd28 100644
--- a/src/sage/sets/primes.py
+++ b/src/sage/sets/primes.py
@@ -877,17 +877,19 @@ class Primes(Set_generic, UniqueRepresentation):
         """
         if other is ZZ:
             return self
-        if not isinstance(other, Primes):
-            raise NotImplementedError("boolean operations are only implemented with other sets of prime numbers")
-        modulus = self._modulus.lcm(other._modulus)
-        classes = [c for c in range(modulus)
-                   if (c % self._modulus in self._classes
-                   and c % other._modulus in other._classes)]
-        exceptions = {x: b for x, b in self._exceptions.items()
-                      if not b or x in other}
-        exceptions.update((x, b) for x, b in other._exceptions.items()
-                          if not b or x in self)
-        return Primes(modulus, classes, exceptions)
+        if isinstance(other, Primes):
+            modulus = self._modulus.lcm(other._modulus)
+            classes = [c for c in range(modulus)
+                       if (c % self._modulus in self._classes
+                       and c % other._modulus in other._classes)]
+            exceptions = {x: b for x, b in self._exceptions.items()
+                          if not b or x in other}
+            exceptions.update((x, b) for x, b in other._exceptions.items()
+                              if not b or x in self)
+            return Primes(modulus, classes, exceptions)
+
+        # assume that other is a finite iterable
+        return Primes(modulus=0, classes=[x for x in other if x in self])

xcaruso · 2025-11-02T23:24:39Z

Ok, I will do it.

mantepse · 2025-11-03T09:50:29Z

src/sage/sets/primes.py

+        elif self.is_finite():
+            modulus = 1
+            classes = []
+            exceptions = {x: True for x in self if x in other}
+        else:
+            # we try to enumerate the elements of "other"
+            modulus = 1
+            classes = []
+            exceptions = {x: True for x in list(other) if x in self}


Are you betting on self being smaller than other if self.is_finite? Personally, I wouldn't, but there are problems in either case:

x in other consumes the iterator, so the following might be surprising:

sage: P = Primes(modulus=0, classes=range(30)) sage: P.intersection(range(10)) Finite set of prime numbers: 2, 3, 5, 7 sage: P.intersection(reversed(range(10))) Finite set of prime numbers: 2

list(other) wastes of time and memory

It may make sense to do a special case for range (untested - I guess one could even accomodate step). This will only make a difference if other.stop is large, I don't know whether this occurs in practice.

Suggested change

elif self.is_finite():

modulus = 1

classes = []

exceptions = {x: True for x in self if x in other}

else:

# we try to enumerate the elements of "other"

modulus = 1

classes = []

exceptions = {x: True for x in list(other) if x in self}

elif isinstance(other, range) and other.step == 1:

modulus = 1

classes = []

exceptions = {}

x = other.start - 1

while True:

x = self.next(x)

if x >= other.stop:

break

exceptions[x] = True

else:

# we try to enumerate the elements of "other"

modulus = 1

classes = []

exceptions = {x: True for x in other if x in self}

The simpler suggestion is:

Suggested change

elif self.is_finite():

modulus = 1

classes = []

exceptions = {x: True for x in self if x in other}

else:

# we try to enumerate the elements of "other"

modulus = 1

classes = []

exceptions = {x: True for x in list(other) if x in self}

else:

# we try to enumerate the elements of "other"

modulus = 1

classes = []

exceptions = {x: True for x in other if x in self}

Are you betting on self being smaller than other if self.is_finite?

No, not necessarily.
But if self is finite and other is infinite, we can compute the intersection; with your suggestions, it will loop forever, no?

* `x in other` consumes the iterator, so the following might be surprising:

It is one reason why I used list(other); the other one was that I get an error message if other is infinite.
I agree nonetheless that this is less efficient and uses more memory.

* It may make sense to do a special case for range (untested - I guess one could even accomodate `step`). This will only make a difference if `other.stop` is large, I don't know whether this occurs in practice.

If other.stop is large and self is finite. Which is already covered by my proposal since I first check if self is finite.

If other is an infinite iterator and x does not appear in other then x in other will also loop forever.

Of course, there are interesting special cases of iterables, most importantly those that implement __contains__.

So, taking into account the above, here is my suggestion for intersection. I think the special case for range objects should generalize to arbitrary step, but I don't have enough patience right now.

index 8f81bc11995..a72ff85f1d0 100644 --- a/src/sage/sets/primes.py +++ b/src/sage/sets/primes.py @@ -20,6 +20,7 @@ AUTHORS: # **************************************************************************** from sage.rings.integer_ring import ZZ +from sage.rings.semirings.non_negative_integer_semiring import NN from sage.rings.infinity import infinity from .set import Set_generic from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets @@ -842,16 +843,22 @@ class Primes(Set_generic, UniqueRepresentation): sage: P.intersection(ZZ) == P True + sage: P.intersection(NN) == P + True sage: P.intersection(RR) Traceback (most recent call last): ... NotImplementedError: object does not support iteration + sage: P = Primes(modulus=0, classes=range(30)) + sage: P.intersection(reversed([13, 7, 11])) + Finite set of prime numbers: 11, 13, 17, 19 + .. SEEALSO:: :meth:`complement_in_primes`, :meth:`union` """ - if other is ZZ: + if other is ZZ or other is NN: return self if isinstance(other, Primes): modulus = self._modulus.lcm(other._modulus) @@ -862,15 +869,25 @@ class Primes(Set_generic, UniqueRepresentation): if not b or x in other} exceptions.update((x, b) for x, b in other._exceptions.items() if not b or x in self) - elif self.is_finite(): - modulus = 1 - classes = [] - exceptions = {x: True for x in self if x in other} else: - # we try to enumerate the elements of "other" modulus = 1 classes = [] - exceptions = {x: True for x in list(other) if x in self} + if isinstance(other, range) and other.step == 1: + exceptions = {} + x = other.start - 1 + while True: + x = self.next(x) + if x >= other.stop: + break + exceptions[x] = True + elif self.is_finite() and hasattr(other, "__contains__"): + # this would not work reliably if ``other`` does not + # implement ``__contains__``, because ``x in other`` + # then consumes ``other`` + exceptions = {x: True for x in self if x in other} + else: + # if other is infinite this will loop forever + exceptions = {x: True for x in other if x in self} return Primes(modulus, classes, exceptions) def union(self, other):

OK.
I just added an additional check to avoid infinite loops if we know in advance that other is not finite.

mantepse

cool!

xcaruso · 2025-11-04T10:53:10Z

Thanks!

@FloFuer

sagemathgh-41122: Subsets of primes defined by congruence conditions We implement subsets of primes given by congruence conditions and operations on those sets. Here is a short demo. ``` sage: P = Primes(modulus=4); P Set of prime numbers congruent to 1 modulo 4: 5, 13, 17, 29, ... sage: P[:10] [5, 13, 17, 29, 37, 41, 53, 61, 73, 89] sage: P.next(500) 509 sage: for p in P: ....: if p > 50: break ....: print(p) 5 13 17 29 37 41 ``` ``` sage: P = Primes(modulus=4); P Set of prime numbers congruent to 1 modulo 4: 5, 13, 17, 29, ... sage: Q = Primes(modulus=4, classes=[3]) Set of prime numbers congruent to 3 modulo 4: 3, 7, 11, 19, ... sage: PQ = P.union(Q) sage: PQ Set of all prime numbers with 2 excluded: 3, 5, 7, 11, ... sage: PQ == Primes().exclude(2) True ``` Cc @FloFuer ### 📝 Checklist  - [x] The title is concise and informative. - [x] The description explains in detail what this PR is about. - [ ] I have linked a relevant issue or discussion. - [x] I have created tests covering the changes. - [x] I have updated the documentation and checked the documentation preview. URL: sagemath#41122 Reported by: Xavier Caruso Reviewer(s): Martin Rubey, Xavier Caruso

initial implementation with documentation

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xcaruso mentioned this pull request Nov 1, 2025

Algebraic and modular properties of hypergeometric functions #41113

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fix typos in documentation

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Martin's suggestions

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Co-authored-by: Martin Rubey <[email protected]>

add INPUT section + allow modulus=0

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typo

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no longer use classes=[] to create finite set of primes

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Update src/sage/sets/primes.py

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Co-authored-by: Martin Rubey <[email protected]>

Update src/sage/sets/primes.py

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Co-authored-by: Martin Rubey <[email protected]>

Update src/sage/sets/primes.py

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replicate changes in union to intersection

243811d

almost equality

1157434

uniformize notation for exceptions: (c, v) -> (x, b)

39b048b

needs pari

09e14fa

use the dict constructor

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Xavier Caruso added 3 commits November 2, 2025 08:01

add getters

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missing doctest

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method in_range

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Update src/sage/sets/primes.py

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Co-authored-by: Martin Rubey <[email protected]>

Update src/sage/sets/primes.py

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supset -> superset

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Merge branch 'primes_congruences' of github.com:xcaruso/sage into pri…

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…mes_congruences

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typos and rephrasing

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improve intersection/union and remove in_range

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improve intersection and union

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Subsets of primes defined by congruence conditions #41122

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