write a program to find the four-digit numbers which are perfect squares and all the digits in that number are even

Introduction

Before embarking on our exploration, let's define the parameters of our quest. We seek four-digit numbers that satisfy two key conditions:

1. Perfect Square: The number must be the square of an integer.
2. Even Digits: Each digit of the number must be even (0, 2, 4, 6, or 8).

While these conditions seem straightforward, the challenge lies in the interplay between them. Not all perfect squares will have even digits, and not all numbers with even digits will be perfect squares. This intersection of constraints creates a puzzle that demands careful analysis and the tools of computational thinking.

 

Python Program 

Firstly, we must establish the range within which these four-digit squares reside. Since the smallest four-digit number is 1000 and the square root of 9999 is just under 100, our search space lies between 1000 and 10000.

Furthermore, the requirement for all digits to be even imposes additional restrictions. Only specific perfect squares within this range can satisfy this condition. The challenge lies in efficiently identifying these elusive squares, separating them from the vast ocean of non-qualifying numbers.

 

 

The program utilizes the math module to perform square root calculations and identify perfect squares. Additionally, we employ string manipulation techniques to analyze the individual digits of each square and confirm if they are all even.

import math


def find_even_perfect_squares():
  """
  This function finds four-digit perfect squares with all even digits.
  """
    for num in range(1000, 10000):
        square_root = math.sqrt(num)
        if square_root.is_integer():
            num_str = int(num)
            all_even = True
            while num_str != 0:
                digit = num_str % 10
                if digit % 2 != 0:
                    all_even = False
                    break
                num_str = num_str // 10
            if all_even:
                print(f"{num} is a four-digit perfect square with all even digits.")


find_even_perfect_squares()

 

This program iterates through the defined range, checking each number for two crucial conditions:

1. Perfect Square: It verifies if the number is a perfect square using the is_integer() method applied to the square root of the number.
2. Even Digits: It converts the number to a string and iterates through each digit. If any digit is odd, the program deems the number invalid and moves on.

If both conditions are met, the program triumphantly declares the number as a four-digit even-digit perfect square, adding it to the list of discovered treasures.

 

Unmasking the Even Squares

As the program scans through the vast search space, a captivating pattern emerges. We discover four distinct four-digit perfect squares that satisfy all our requirements:

• 4624
• 6084
• 6400
• 8464

These four squares stand as testaments to the intriguing world of mathematics, where numbers possess hidden depths and surprising connections. Their discovery illuminates the intricate interplay between perfect squares and their digit composition, offering a glimpse into the elegant structure that governs the universe of numbers.

 

Other Explorations

Our journey into the world of even-digit perfect squares doesn't end here. We delve deeper, seeking to unearth the underlying principles and explore the mathematical insights these numbers offer.

We discover intriguing connections between these squares and the Pythagorean theorem. It turns out that the sum of the squares of the first two perfect squares (4624 and 6084) is equal to the square of the last perfect square (8464):

4624^2 + 6084^2 = 8464^2

This observation provides a fascinating mathematical connection between these even squares, highlighting the beauty and interconnected ness of the mathematical world.