How much of the standard proof of the fundamental theorem of arithmetic follows from general tricks that can be applied all over the place and how much do you actually have to remember? At first it may seem as though you have to remember quite a bit: there is a non-obvious sequence of lemmas, starting with Bézout’s theorem, continuing with the clever proof that if p|ab then either p|a or p|b, bumping that up to a proof for bigger products, and eventually deducing the theorem itself.
But what if one were simply asked to come up with a proof? Would there be any chance of discovering that sequence of lemmas? I maintain that there would — if, that is, you are aware of certain general tricks.
Let’s imagine, then, that we don’t know the proof and are trying to work it out. I’ll split the whole process up into a number of steps. I’ll precede the description of each step by a slogan that more or less generates the argument.
1. State the problem carefully and give names to things.