> So I guess if you're comfortable with modular arithmetic, you can fairly consider this an obvious proof. It relies on another result about primes, but it's very common that one result makes another result easy, and a blanket disallowal of that approach leaves you saying that proving anything is as tricky and non-obvious as proving, proving, that 2+2=4.
But Gowers' point is that FTA relies on a deep fact that is non trivial (as pointed out and proven elsewhere on this thread, the correctness of Euclidean division: https://en.wikipedia.org/wiki/Euclidean_division#Proof). You can sidestep this by using other lemmas in arithmetic, but at some point everything is resting on this one deep fact.
Also, it's not like proving that 2+2=4, which follows trivially from Peano's axioms and the definitions of 2, 4, and +. If something is nontrivial to prove, it's usually because there is something deeper going on beneath. In this case, the fact that Euclidean division works for the integers, but not for other number systems (e.g., Z[sqrt(-5)]), is what makes it deep and interesting.
But Gowers' point is that FTA relies on a deep fact that is non trivial (as pointed out and proven elsewhere on this thread, the correctness of Euclidean division: https://en.wikipedia.org/wiki/Euclidean_division#Proof). You can sidestep this by using other lemmas in arithmetic, but at some point everything is resting on this one deep fact.
Also, it's not like proving that 2+2=4, which follows trivially from Peano's axioms and the definitions of 2, 4, and +. If something is nontrivial to prove, it's usually because there is something deeper going on beneath. In this case, the fact that Euclidean division works for the integers, but not for other number systems (e.g., Z[sqrt(-5)]), is what makes it deep and interesting.