pi

Is it possible for the equation to end? What would happen if mathematicians discover it (or find a pattern)?

They might find a pattern.  What would happen?  Nothing.

There is no reason in principle why it might not turn out that after the quadrillionth digit, the digits of pi are 3733733333373333333333333333333333337333... .  (If this did happen, it would be utterly shocking to mathematicians, although perhaps not to anyone else.)  Pi is known to be irrational, but this only rules out "periodic" patterns, where a fixed sequence of digits repeats exactly the same, over and over.