Echo Perfect Square

October 2, 2025

Question

For a given positive integer x, define its echo to be x printed twice in a row. For example the echo of 23 would be 2323. Can you find an integer with an echo that is a perfect square? If you can, explain what qualifies an integer to work. If you believe none exist, provide a proof. This does not need to be completely rigorous, but should follow the framework of a mathematical proof; a simple justification or trial of a few cases will not suffice.

Solution

I'm working on getting the proof up for this (I've solved it, it's just annoying to format and upload a proof; if you've seen my handwriting you'll understand why this is necessary), but it was interesting. I was severely sick with mold poisoning, which I didn't know at the time, so this took me quite a while to think up. If you want to solve it, think of the factors two numbers would have to independently have versus share to comprise a square number, think of how to represent an echo in terms of its echee(?), and consider "number of integers" reasoning.