Start with a perfect square, denoted as a positive integer **n**.
The root of this square is **k**, another positive integer.
Thus **n = k^2**
Let **t** = totient(**n**)

Is there a way to prove there is no such number? I know that Fermat Primes have totient values that are squares, but what about any perfect squares who have a perfect square as their totient value?

And more generally, expanding this concept with an additional restriction, can we say that a totient value (derived from a square number) will not be part of a Pythagorean Triple?

**n** = **t** + **a**

Where **a** is another integer, and is the difference between **n** and its totient value.

**a** = **n** - **t**

In other words, can **n**, **t**, and **a** all be perfect squares?

Furthermore, if this is disprovable, then can we also disprove the case of Pythagorean Quadruples?
Where perfect squares must sum together to equal the totient value of a perfect square? Does removing the restriction of it being a Pythagorean Triple/Quadruple, and allowing **a** to be non-square make this possible?

I am not a mathematician. I'm an Electrical Engineer, and have been exploring the world of Number Theory. I find it fascinating and have posed this question for myself, as I have not seen an answer. I believe it's possible I've missed something simple here and there is an "easy" way to show that such things cannot be. I have my computer checking by brute force if any totients of **n** are also square, and got up to the first 10,000 perfect squares as having no totient number that is a perfect square.

Any insight is appreciated, and I am sorry for the lack of formatting and formal means of formulating my question. I hope this question is clear. I ultimately want to know if there is any relationship between the Pythagorean Triples/Quadruples(or higher sets) to Euler's Totient Function. Wondering if by "coincidence" there is such a number, or if it is impossible (and why). Thank you for taking the time to think about this with me. This is a related question, but is not my question: Pythagorean triples that "survive" Euler's totient function