From e783e1473aaf19c80a2e8fba4059baa949853521 Mon Sep 17 00:00:00 2001
From: Bibhukesh Medhi <2102508728@svyasa-sas.edu.in>
Date: Wed, 22 Oct 2025 13:35:33 +0530
Subject: [PATCH 1/3] added DSA problem of rare_number

---
 maths/rare_number.py | 107 +++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 107 insertions(+)
 create mode 100644 maths/rare_number.py

diff --git a/maths/rare_number.py b/maths/rare_number.py
new file mode 100644
index 000000000000..92051aba7a83
--- /dev/null
+++ b/maths/rare_number.py
@@ -0,0 +1,107 @@
+"""
+Rare numbers are special non-palindromic numbers N such that both N + rev(N)
+and N - rev(N) are perfect squares, where rev(N) is the reverse of the digits
+of N.
+
+This module provides functions to check and generate rare numbers in a given
+range. Rare numbers are useful in number theory and pattern-based algorithm
+analysis.
+
+For more information about rare numbers, refer to:
+[https://www.geeksforgeeks.org/dsa/rare-numbers/](https://www.geeksforgeeks.org/dsa/rare-numbers/)
+"""
+
+import math
+
+
+def rare_numbers(start: int, end: int) -> list[int]:
+    """
+    Find all rare numbers between the given start and end range (inclusive).
+
+    A rare number N satisfies both:
+    1. N + rev(N) is a perfect square.
+    2. N - rev(N) is a perfect square.
+    where rev(N) is the reverse of the digits of N,
+    and N must not be a palindrome.
+
+    Args:
+        start (int): The lower bound of the range (inclusive).
+        end (int): The upper bound of the range (inclusive).
+
+    Returns:
+        list[int]: A list of rare numbers within the specified range.
+
+    Raises:
+        ValueError: If start or end is negative, or start > end.
+
+    Examples:
+    >>> rare_numbers(-1, 100)
+    Traceback (most recent call last):
+        ...
+    ValueError: Range limits must be non-negative and start <= end
+    >>> rare_numbers(1, 100)
+    []
+    >>> rare_numbers(1, 1000)
+    [65]
+    """
+    if start < 0 or end < 0 or start > end:
+        raise ValueError("Range limits must be non-negative and start <= end")
+
+    rares = []
+    for n in range(start, end + 1):
+        rev_n = _reverse_number(n)
+        if n == rev_n:
+            continue  # skip palindromes
+        if _is_perfect_square(n + rev_n) and _is_perfect_square(abs(n - rev_n)):
+            rares.append(n)
+    return rares
+
+
+def _reverse_number(num: int) -> int:
+    """
+    Reverse the digits of a given integer.
+
+    Args:
+        num (int): The integer to reverse.
+
+    Returns:
+        int: The reversed integer.
+
+    Examples:
+    >>> _reverse_number(123)
+    321
+    >>> _reverse_number(400)
+    4
+    """
+    rev = 0
+    while num > 0:
+        rev = rev * 10 + num % 10
+        num //= 10
+    return rev
+
+
+def _is_perfect_square(n: int) -> bool:
+    """
+    Check if a number is a perfect square.
+
+    Args:
+        n (int): The number to check.
+
+    Returns:
+        bool: True if n is a perfect square, False otherwise.
+
+    Examples:
+    >>> _is_perfect_square(16)
+    True
+    >>> _is_perfect_square(20)
+    False
+    """
+    if n < 0:
+        return False
+    root = int(math.isqrt(n))
+    return root * root == n
+
+
+if __name__ == "__main__":
+    import doctest
+    doctest.testmod()

From f9680cf1d65c9ed31f23a31167f8420d2b855f71 Mon Sep 17 00:00:00 2001
From: "pre-commit-ci[bot]"
 <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Date: Wed, 22 Oct 2025 08:29:02 +0000
Subject: [PATCH 2/3] [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci
---
 maths/rare_number.py | 1 +
 1 file changed, 1 insertion(+)

diff --git a/maths/rare_number.py b/maths/rare_number.py
index 92051aba7a83..5b64549c9958 100644
--- a/maths/rare_number.py
+++ b/maths/rare_number.py
@@ -104,4 +104,5 @@ def _is_perfect_square(n: int) -> bool:
 
 if __name__ == "__main__":
     import doctest
+
     doctest.testmod()

From 5523aec2e8c1a51dfa07cafcced5d9cf4fcc4dfb Mon Sep 17 00:00:00 2001
From: bibhukesh22 <2102508728@svyasa-sas.edu.in>
Date: Thu, 23 Oct 2025 16:30:58 +0530
Subject: [PATCH 3/3] Fix condition for identifying rare numbers

---
 maths/rare_number.py | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/maths/rare_number.py b/maths/rare_number.py
index 5b64549c9958..38693acdd251 100644
--- a/maths/rare_number.py
+++ b/maths/rare_number.py
@@ -52,7 +52,7 @@ def rare_numbers(start: int, end: int) -> list[int]:
         rev_n = _reverse_number(n)
         if n == rev_n:
             continue  # skip palindromes
-        if _is_perfect_square(n + rev_n) and _is_perfect_square(abs(n - rev_n)):
+     if n - rev_n > 0 and _is_perfect_square(n + rev_n) and _is_perfect_square(n - rev_n):
             rares.append(n)
     return rares
 
@@ -98,7 +98,7 @@ def _is_perfect_square(n: int) -> bool:
     """
     if n < 0:
         return False
-    root = int(math.isqrt(n))
+    root = math.isqrt(n)
     return root * root == n