This is sort of how Fermat's last theorem was solved by Andrew Wiles (forgive me if I misrepresent this proof, my math is a few years rusty) - by creating a different kind of representation of elliptic curves, it was possible to compare them to modular forms in a way that created a contradiction that proved the theorem correct.

Well some numbers expose patterns when written as a continued fraction. In particular e becomes pretty regular.

You can modify the continued fraction slightly to make pi regular as well, but the normal continued fraction sequence doesn't give much of an insight. Other than the fact that 3 + 1/(7 + 1/16)) is a damn good approximation (7 digits, pretty good for something that can be written using only 4 digits total: [3;7,16]).

Phi/golden ratio also has a cool continued fraction sequence...it's only 1's all the way down

Square roots in general have periodic patterns. Which isn't too surprising, something like z = a/(b+cz) is pretty much a quadratic equation after all.

But phi is indeed especially interesting because of what its sequence implies for rational approximations of phi.

Larger integers in continued fractions mean you get 'more information' out of the limb. That means not only is Phi "1s all the way down" it is the continued fraction that converges the slowest. If you've ever used the iterated matrix product (which is a specific edge-case of the algorithm to convert continued fractions to decimals), you'll know how slow it is!

One example is Ulam spiral: https://en.wikipedia.org/wiki/Ulam_spiral

I guess the Ulam spiral is more an artefact of the cartesian plane than an "hidden pattern": https://www.youtube.com/watch?v=EK32jo7i5LQ

maybe do it in another base rather than base 10. Just because we have 10 fingers does not mean a god does.