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? 

You can think of this probabilistically. Let's say I want to test a hypothesis (you can think of any real world example, such as drug testings) and I want to make sure that I'm right in at least $p\% > 0$ cases to be satisfied. But, if I have some statement about natural numbers (or some infinite subset of natural numbers) and I check it to be true for all the numbers lesser than some large $n$, I am still $0\%$ sure my statement is correct since $\frac n \infty = 0$. No matter how many different numbers you check, ultimately you didn't catch even a tiny fraction. That is why we use deductive reasoning in mathematics, not empirical experiments.
