Consider a number in its decimal representation that begins with an odd number of consecutive digits of 4, followed by a single digit of 1. An example of such a number would be 41, 4441, or any similar pattern extending with 4s. My question is: Can such a number ever be a perfect square?
To clarify, the numbers we're considering take the form 44 … 41 44…41, where the number of 4's is odd, and it's terminated by a single 1.
Here are the specific points I'm curious about:
- Is there a mathematical approach or theorem that directly addresses the properties of numbers with specific digit patterns in relation to being perfect squares?
- Could modular arithmetic or any form of number theory provide insight into proving or disproving the possibility of such a number being a perfect square?
I've attempted some preliminary analysis, including playing around with smaller cases and considering the last digits of square numbers, but haven't reached a conclusive answer. Any guidance, references, or insights into how to approach this problem would be greatly appreciated.
Thank you in advance for your time and assistance!