> I'm also going to have to strongly disagree with you in regards to Euler's proofs. Analysis as a discipline is far more mature than it was in Euler's time, but Euler still clearly proved the things we attribute to him. I'm not really sure what your expectation is here - we have more sophisticated means of articulating proofs, and likewise much more mathematical scaffolding to support new theories. But Euler's work is rigorous.
What about the Euler product formula and using it to prove the infinitude of primes? At best it has gaps that can be easily filled, but it wouldn't meet modern standards of rigor. See http://www.math.harvard.edu/~elkies/M259.06/euler.pdf (the best source I can find on my phone)