python · skirpichev · May 22, 2026 · May 23, 2026 · May 23, 2026 · May 23, 2026
diff --git a/Doc/library/math.integer.rst b/Doc/library/math.integer.rst
@@ -62,6 +62,12 @@ computed exactly and are integers.
       :trim:
 
 
+.. function:: isqrt_rem(n, /)
+
+    Return a pair of values (s,t) such that s=isqrt(n) and t=n-s*s.
+    The remainder *t* will be zero, if *n* is a perfect square.
+
+
 .. function:: lcm(*integers)
 
    Return the least common multiple of the specified integer arguments.

@@ -86,14 +86,20 @@ New modules
 Improved modules
 ================
 
-
 gzip
 ----
 
 * :func:`gzip.open` now accepts an optional argument ``mtime``
   which is passed on to the constructor of the :class:`~gzip.GzipFile` class.
   (Contributed by Marin Misur in :gh:`91372`.)
 
+math.integer
+------------
+
+* Add :func:`math.integer.isqrt_rem` to compute integer square root with
+  a remainder.
+  (Contributed by Sergey B Kirpichev in :gh:`90345`.)
+
 os
 --
 

diff --git a/Lib/test/test_math_integer.py b/Lib/test/test_math_integer.py
@@ -2,6 +2,7 @@
 from fractions import Fraction
 import unittest
 from test import support
+from math.integer import isqrt_rem
 
 
 class IntSubclass(int):
@@ -249,6 +250,44 @@ def test_isqrt_huge(self, size):
         self.assertEqual(w.bit_length(), size // 2 + 1)
         self.assertEqual(w.bit_count(), 1)
 
+    def test_isqrt_rem(self):
+        test_values = (
+            list(range(1000))
+            + list(range(10**6 - 1000, 10**6 + 1000))
+            + [2**e + i for e in range(60, 200) for i in range(-40, 40)]
+            + [3**9999, 10**5001]
+        )
+        for value in test_values:
+            with self.subTest(value=value):
+                root, rem = isqrt_rem(value)
+                self.assertIs(type(root), int)
+                self.assertLessEqual(root*root, value)
+                self.assertLess(value, (root+1)*(root+1))
+                self.assertIs(type(rem), int)
+                self.assertEqual(rem, value - root*root)
+
+        # Negative values
+        with self.assertRaises(ValueError):
+            isqrt_rem(-1)
+
+        # Integer-like things
+        self.assertEqual(isqrt_rem(True), (1, 0))
+        self.assertEqual(isqrt_rem(False), (0, 0))
+        self.assertEqual(isqrt_rem(MyIndexable(1729)), (41, 48))
+
+        with self.assertRaises(ValueError):
+            isqrt_rem(MyIndexable(-3))
+
+        # Non-integer-like things
+        bad_values = [
+            3.5, "a string", Decimal("3.5"), 3.5j,
+            100.0, -4.0,
+        ]
+        for value in bad_values:
+            with self.subTest(value=value):
+                with self.assertRaises(TypeError):
+                    isqrt_rem(value)
+
     def test_perm(self):
         perm = self.module.perm
         factorial = self.module.factorial

diff --git a/Misc/NEWS.d/next/Library/2026-05-22-11-32-21.gh-issue-90345.Z9QjnG.rst b/Misc/NEWS.d/next/Library/2026-05-22-11-32-21.gh-issue-90345.Z9QjnG.rst
@@ -0,0 +1 @@
+Add :func:`math.integer.isqrt_rem`.
diff --git a/Modules/clinic/mathintegermodule.c.h b/Modules/clinic/mathintegermodule.c.h
diff --git a/Modules/mathintegermodule.c b/Modules/mathintegermodule.c
@@ -341,18 +341,8 @@ _approximate_isqrt(uint64_t n)
     return (u << 15) + (uint32_t)((n >> 17) / u);
 }
 
-/*[clinic input]
-math.integer.isqrt
-
-    n: object
-    /
-
-Return the integer part of the square root of the input.
-[clinic start generated code]*/
-
 static PyObject *
-math_integer_isqrt(PyObject *module, PyObject *n)
-/*[clinic end generated code: output=551031e41a0f5d9e input=921ddd9853133d8d]*/
+_isqrt_rem(PyObject *n, PyObject **rem)
 {
     int a_too_large, c_bit_length;
     int64_t c, d;
@@ -373,6 +363,9 @@ math_integer_isqrt(PyObject *module, PyObject *n)
     }
     if (_PyLong_IsZero((PyLongObject *)n)) {
         Py_DECREF(n);
+        if (rem) {
+            *rem = PyLong_FromLong(0);
+        }
         return PyLong_FromLong(0);
     }
 
@@ -392,7 +385,15 @@ math_integer_isqrt(PyObject *module, PyObject *n)
             return NULL;
         }
         u = _approximate_isqrt(m << 2*shift) >> shift;
-        u -= (uint64_t)u * u > m;
+        uint64_t sq = (uint64_t)u * u;
+        u -= sq > m;
+        if (rem) {
+            if (sq > m) {
+                sq -= 2*(uint64_t)u + 1;
+            }
+            m -= sq;
+            *rem = PyLong_FromUnsignedLongLong(m);
+        }
         return PyLong_FromUnsignedLong(u);
     }
 
@@ -460,14 +461,37 @@ math_integer_isqrt(PyObject *module, PyObject *n)
         goto error;
     }
     a_too_large = PyObject_RichCompareBool(n, b, Py_LT);
-    Py_DECREF(b);
     if (a_too_large == -1) {
+        Py_DECREF(b);
         goto error;
     }
 
     if (a_too_large) {
+        if (rem) {
+            PyObject *tmp = PyNumber_Add(b, _PyLong_GetOne());
+
+            if (tmp == NULL) {
+                Py_DECREF(b);
+                goto error;
+            }
+            Py_SETREF(b, tmp);
+            tmp = PyNumber_Add(a, a);
+            if (tmp == NULL) {
+                Py_DECREF(b);
+                goto error;
+            }
+            Py_SETREF(b, PyNumber_Subtract(b, tmp));
+            Py_DECREF(tmp);
+        }
         Py_SETREF(a, PyNumber_Subtract(a, _PyLong_GetOne()));
     }
+    if (!rem) {
+        Py_DECREF(b);
+    }
+    else {
+        Py_SETREF(b, PyNumber_Subtract(n, b));
+        *rem = b;
+    }
     Py_DECREF(n);
     return a;
 
@@ -478,6 +502,49 @@ math_integer_isqrt(PyObject *module, PyObject *n)
 }
 
 
+/*[clinic input]
+math.integer.isqrt
+
+    n: object
+    /
+
+Return the integer part of the square root of the input.
+[clinic start generated code]*/
+
+static PyObject *
+math_integer_isqrt(PyObject *module, PyObject *n)
+/*[clinic end generated code: output=551031e41a0f5d9e input=921ddd9853133d8d]*/
+{
+    return _isqrt_rem(n, NULL);
+}
+
+/*[clinic input]
+math.integer.isqrt_rem
+
+    n: object
+    /
+
+Return a pair of values (s,t) such that s=isqrt(n) and t=n-s*s.
+[clinic start generated code]*/
+
+static PyObject *
+math_integer_isqrt_rem(PyObject *module, PyObject *n)
+/*[clinic end generated code: output=b17d11479d08cdc4 input=7ed2dd870818d2bb]*/
+{
+    PyObject *rem = NULL;
+    PyObject *root = _isqrt_rem(n, &rem);
+    PyObject *res = NULL;
+
+    if (root && rem) {
+        res = PyTuple_Pack(2, root, rem);
+    }
+    Py_XDECREF(root);
+    Py_XDECREF(rem);
+    return res;
+    return NULL;
+}
+
+
 static unsigned long
 count_set_bits(unsigned long n)
 {
@@ -1231,6 +1298,7 @@ static PyMethodDef math_integer_methods[] = {
     MATH_INTEGER_FACTORIAL_METHODDEF
     MATH_INTEGER_GCD_METHODDEF
     MATH_INTEGER_ISQRT_METHODDEF
+    MATH_INTEGER_ISQRT_REM_METHODDEF
     MATH_INTEGER_LCM_METHODDEF
     MATH_INTEGER_PERM_METHODDEF
     {NULL,              NULL}           /* sentinel */