What equations, or results about equations, generalize in interesting ways from number theory or geometry to more abstract settings? The motivating example for this question was as follows:

Pythagorean triples are one of the first mathematical concepts many people encounter. But is the equation a^{2} + b^{2} = c^{2} of any relevance in more general rings? The rings of square matrices over the integers (or rationals), for example? Rings of polynomials? The ring of integers of a number field - or just the number field itself? How about more abstract settings?

Other equations spring to mind, however. To take a related example, what rings is Fermat's last theorem thought (or known) to generalize to?

What about the equation, x^{2} = x + 1, defining the Golden ratio? Is that of any interest? Or the iterated mapping which generates the Mandelbrot set (say over C^{n×n})?

What use are things like exponentiation, trigonometric functions and π when studying, say, matrices over a complete field? Are they only used for solving differential equations, or are there more number-theory-like applications (as in the relationship between π and the primes, for example)?

Apologies if all this is a bit naive; likewise if it's too much for one question - I'm sure FLT alone would lead to interesting answers. Also, feel free to make this community wiki if that's appropriate: I'm still relatively new to this site, and unsure of etiquette.