Is a pattern proof? Let's say I want a formula that takes any number and makes it into 170, and I come up with a formula that I think does it. If I plug 1 into it, 2 into it, 3 into it, etc. up to a pretty large number and it works, can I call the formula proven? 

Here's a very contrived example: $$f(n) = \Big\lfloor \frac{\pi(n)}{10^7} \Big\rfloor + 170,$$ where $\pi(x)$ is the prime counting function. You'd have to go above $n = 179424672$ to find a counterexample. Of course "$f(n) = 170$ always" is easy enough to disprove if you know that there are infinitely many prime numbers, so it requires very little actual computation to disprove.
