So, I'm reading How To Prove It, by Daniel J. Velleman as I'm on a quest right now to work on proofs everyday. In then introduction, he says that mathematicians look for patterns. When you see a pattern you can call it a conjecture. A conjecture is just an educated guess. So, for example he gives an exercise to make some conjectures about the values of n for which 3^n-1 is prime or the values of n for which (3^n) - (2^n) is prime.
Step one of this is to make a table of results and then look for a pattern:
I made this table using excel. I'm going to run it for more numbers and over the next day or so, I'm going to try to come up with some conjectures about the data and then try to prove the conjectures. Theorems are conjectures that have been proven.