diff --git a/mathics/core/expression.py b/mathics/core/expression.py
index 0be8e99e5..369aed1a3 100644
--- a/mathics/core/expression.py
+++ b/mathics/core/expression.py
@@ -371,6 +371,8 @@ class Expression(BaseElement, NumericOperators, EvalMixin):
     _cache: ExpressionCache | None
     # (head, elements, element_order) of the last element_order built.
     _element_order_cache: tuple[BaseElement, tuple[BaseElement, ...], tuple] | None
+    # Whether no Slot occurs in the expression: see _has_no_slots().
+    _slot_free: bool | None = None
     elements_properties: ElementsProperties | None
     options: dict[str, Any] | None
     pattern_sequence: bool
@@ -614,6 +616,7 @@ def elements(self, values: Iterable):
         # Set to build self.elements_properties on next evaluation()
         self.elements_properties = None
         self._cache = None
+        self._slot_free = None
         self._element_order_cache = None
 
     def equal2(self, rhs: Any) -> bool | None:
@@ -1732,6 +1735,7 @@ def set_head(self, head: Symbol):
             raise TypeError("Attempt to turn an Expression into a ListExpression")
         self._head = head
         self._cache = None
+        self._slot_free = None
         self._element_order_cache = None
 
     def set_element(self, index: int, value):
@@ -2012,6 +2016,28 @@ def replace_vars(self, vars, options=None, in_function=True) -> "Expression":
             return self
         return Expression(head, *new_elements)
 
+    def _has_no_slots(self) -> bool:
+        """
+        True if no Slot or SlotSequence occurs in the expression, heads
+        included. Worked out once and kept with the expression, so that a
+        part shared by every application of a pure function is looked at
+        once; changing the head or the elements forgets it.
+        """
+        flag = self._slot_free
+        if flag is None:
+            head = self._head
+            flag = (
+                head is not SymbolSlot
+                and head is not SymbolSlotSequence
+                and (not isinstance(head, Expression) or head._has_no_slots())
+                and all(
+                    not isinstance(element, Expression) or element._has_no_slots()
+                    for element in self._elements
+                )
+            )
+            self._slot_free = flag
+        return flag
+
     def replace_slots(self, slots, evaluation):
         """
         Replaces Slots (#1, ##, etc) by the corresponding values in `slots`
@@ -2038,6 +2064,10 @@ def replace_slots(self, slots, evaluation):
         elif self._head is SymbolFunction and len(self._elements) == 1:
             # do not replace Slots in nested Functions
             return self
+        if self._has_no_slots():
+            # No slot anywhere below, e.g. a graph passed to a function and
+            # written into the body of a pure function: nothing to walk.
+            return self
         head = self._head.replace_slots(slots, evaluation)
         elements = [
             element.replace_slots(slots, evaluation) for element in self._elements
diff --git a/test/core/test_replace_slots.py b/test/core/test_replace_slots.py
new file mode 100644
index 000000000..777b2ed11
--- /dev/null
+++ b/test/core/test_replace_slots.py
@@ -0,0 +1,68 @@
+# -*- coding: utf-8 -*-
+"""
+Applying a pure function does not walk the parts of its body that have no
+slot in them.
+"""
+
+from test.helper import check_evaluation, session
+
+import pytest
+
+from mathics.core.expression import Expression
+
+
+def test_slot_free_part_is_not_walked(monkeypatch):
+    big = session.evaluate("Table[{i, w -> i}, {i, 300}]")
+    body = session.evaluate("Hold[{#1, Length[x]}]").elements[0]
+    # The body {#1, Length[<big>]}, as when a list is passed to a function
+    # and written into a pure function inside it.
+    body = Expression(
+        body.head, body.elements[0], Expression(body.elements[1].head, big)
+    )
+    calls = [0]
+    original = Expression.replace_slots
+
+    def counting(self, slots, evaluation):
+        calls[0] += 1
+        return original(self, slots, evaluation)
+
+    monkeypatch.setattr(Expression, "replace_slots", counting)
+    for i in range(5):
+        new = body.replace_slots([None, session.evaluate(str(i))], session.evaluation)
+        assert new.elements[1].elements[0] is big
+    # Three calls per application (the body, #1 and Length[big]),
+    # not one for each of the 900 subexpressions of big.
+    assert calls[0] <= 5 * 4, f"{calls[0]} calls for 5 applications"
+
+
+@pytest.mark.parametrize(
+    ("str_expr", "str_expected"),
+    [
+        (
+            "rsData = Range[5]; rsF[l_] := Map[{#, Length[l]} &, {a, b}]; rsF[rsData]",
+            "{{a, 5}, {b, 5}}",
+        ),
+        ("Map[{#, {1, 2}} &, {a, b}]", "{{a, {1, 2}}, {b, {1, 2}}}"),
+        ("Apply[{##, {x}} &, {a, b}]", "{a, b, {x}}"),
+        ("Map[Function[{#, # &}], {1}]", "{{1, #1 &}}"),
+        ("Map[{#, Hold[#]} &, {a}]", "{{a, Hold[a]}}"),
+        ("ClearAll[rsData, rsF]", "Null"),
+    ],
+)
+def test_pure_functions(str_expr, str_expected):
+    check_evaluation(str_expr, str_expected)
+
+
+@pytest.mark.parametrize(
+    ("str_expr", "str_expected"),
+    [
+        # Slots in a compound head, at any depth.
+        ("Map[#[0] &, {Sin, Cos}]", "{0, 1}"),
+        ("Map[{a, {#[0]}} &, {Cos}]", "{{a, {1}}}"),
+        ("Map[f[#][x] &, {1, 2}]", "{f[1][x], f[2][x]}"),
+        ("Map[g[h[#]][x] &, {1}]", "{g[h[1]][x]}"),
+        ("Apply[f[##][y] &, {1, 2}]", "f[1, 2][y]"),
+    ],
+)
+def test_slots_in_compound_heads(str_expr, str_expected):
+    check_evaluation(str_expr, str_expected)