I offered much of the following to McWorter as a comment to a probem relating to a Square Value for the number 2005. There seemed to be the implication that the base was integral, however, within comments at least one other offered a different base.
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I was probably asking for comment for what should properly be addressed within this section of the site.

Essentially my comment was:
I wonder about the philosophical legitimacy of my solution, but within the limited parameters which I have defined for myself, my solution has validity.

If I take the number 4 (base 10) into a modular arithmetical/algebraic environment, eg mod(5), 4 still remains as 4. It is still the square of 2.

Now I think I tread on eggshells:
I am considering 2005 base 10 under the modulus of 2001. 2001 => 0, 2002 => 1, 2003 => 2, 2004 => 3 and 2005 => 4!
Yes, 2 * 2 = 4, and I have a square.

A generality of my approach to this is: 2005 (mod(2001-x^2)) = x * x.

Now before any would like to drag me violently kicking to La Place de L'Etoile to exact penance as a heretic I will offer these considerations:
1. Back to base 10 mod(5). 2 * 2 = 4, but 4 * 4 = 1. [Now here might be another base to work from!]
2. In reality, the highest value that can exist for mod(x) is x-1.
Good piece of trivial exploration? But I guess that I've destroyed my own case! Bonjour Madame Guillotine!
Hey! Sorry, this is probably a Forum matter, but if I get away with playing with squares in a modulus environment, then why not .... primes ... Fermat numbers ... Fibonacci series ...
Some of these instances obvious come up in practical program situations, but philosophically, is this a legitimate tack? Yes, I will take it there. Comments to that end, please go to Forums/General.