I definitely get where you're coming from, but the introduction of Z[sqrt(-5)] was just there to illustrate that proof is hard. Even if you never invoke an exot

I definitely get where you're coming from, but the introduction of Z[sqrt(-5)] was just there to illustrate that proof is hard. Even if you never invoke an exotic example it's hard to have proof that FTA holds. The simplest way to get at that perhaps is to just increase the scope. It's obvious to you that 10 and 15 have unique factorizations. You can compute them and convince yourself that nothing else would work. It's harder to do the same for 9^87654321-1 but you could in principle. Furthermore, you believe it would work (you could, e.g., program a computer to do it and wait a few hours? days? something like that) but only basically because you see no reason for your experience with small numbers to eventually stop. But now let me tell you about the Ackermann function and ask you about A(1000, 1000). Let's be clear, this is just a really large number, but now we're at a point where a computer couldn't write down the factorization in the lifetime of the universe. Are you still sure it's obvious? It might at this point become more clear that "I see no reason for my intuition about small numbers to stop" is weaker than "I know this always works".