Check if a Number is a Perfect Square

Determining whether a given number is a perfect square is a common problem in programming, with applications ranging from basic number theory to more complex algorithms. A perfect square is a number that can be obtained by squaring an integer (e.g., 9 is a perfect square because 3*3 = 9). This post explores several efficient ways to check for perfect squares in Python, providing clear code examples and explanations for each method.

Method 1: Using the math.isqrt() function (Python 3.8+)

Python 3.8 introduced the math.isqrt() function, which provides the integer square root of a number. This is arguably the most efficient and straightforward method for checking if a number is a perfect square. It returns the floor of the square root. If the square of this integer square root equals the original number, then it’s a perfect square.

import math

def is_perfect_square_isqrt(n):
  """Checks if n is a perfect square using math.isqrt().

  Args:
    n: The number to check.  Must be a non-negative integer.

  Returns:
    True if n is a perfect square, False otherwise.
  """
  if n < 0:
    return False  # Handle negative numbers
  root = math.isqrt(n)
  return root * root == n

print(is_perfect_square_isqrt(16))  # Output: True
print(is_perfect_square_isqrt(17))  # Output: False
print(is_perfect_square_isqrt(0))   # Output: True
print(is_perfect_square_isqrt(-4))  # Output: False

Method 2: Using the ** operator and type checking

This method leverages Python’s exponentiation operator (**) to calculate the square root and then checks if the result is an integer. We use isinstance() to ensure the result is an integer. This method is less efficient than math.isqrt() but is a good alternative for older Python versions.

def is_perfect_square_exponent(n):
  """Checks if n is a perfect square using the exponentiation operator.

  Args:
    n: The number to check. Must be a non-negative number.

  Returns:
    True if n is a perfect square, False otherwise.
  """
  if n < 0:
    return False
  root = n**0.5
  return isinstance(root, int)

print(is_perfect_square_exponent(25)) # Output: True
print(is_perfect_square_exponent(26)) # Output: False

Method 3: Binary Search (for educational purposes)

While less efficient than the previous methods, a binary search approach demonstrates a different algorithmic strategy. This is primarily for educational purposes, showcasing a different way to solve the problem.

def is_perfect_square_binary_search(n):
  """Checks if n is a perfect square using binary search.

  Args:
    n: The number to check. Must be a non-negative integer.

  Returns:
    True if n is a perfect square, False otherwise.
  """
  if n < 0:
    return False
  low, high = 0, n
  while low <= high:
    mid = (low + high) // 2
    square = mid * mid
    if square == n:
      return True
    elif square < n:
      low = mid + 1
    else:
      high = mid - 1
  return False

print(is_perfect_square_binary_search(64)) # Output: True
print(is_perfect_square_binary_search(65)) # Output: False

Each method offers a slightly different trade-off between readability, efficiency, and compatibility with various Python versions. For optimal performance, especially with larger numbers, math.isqrt() is recommended. The other methods provide valuable insights into alternative approaches and are useful for understanding underlying concepts.