TITLE: What are some elementary results (number theory) using theorems that went long-unproven?
QUESTION [8 upvotes]: Firstly, I do not mind if there are examples from fields other than number theory! This was just the first field, and where I think the richest examples, may come from.
Now to elaborate on the title, if only a little: Are there any elementary number theory proofs that have since been made using theorems (such as Fermat's Last Theorem or Goldbach's Weak Theorem $^{1}$) that went unproven for a long period of time? Or, in general, elementary proofs using theorems that required an enormous amount of machinery (algebraic geometry, modularity theorem, etc.)?
I wouldn't mind seeing elementary proofs of a result that had been already proven. But new proofs would also be very intriguing. Part of the reason for thinking in this way is this: I wonder where people in the previous era of mathematics would have looked next had they been able to use these theorems. Thanks for the help.

$^1$ this is a theorem now, right?
Edit: Added the big-list tag and bounty in hopes of more answers.

REPLY [5 votes]: There's also a well known overpowered proof of the infinitude of primes:
If $P$ is the set of primes, the Euler product formula tells us
$$\prod\limits_{p\in P}\frac{1}{1-\frac{1}{p^2}}=\zeta(2)=\frac{\pi^2}{6}$$
However, $\pi$ is transendental, so $|P|$ cannot be finite (otherwise the product would be rational).