TITLE: Elementary proof of Zsigmondy's theorem
QUESTION [32 upvotes]: I've been writing a not-so-short introduction to elementary number theory, supplying proofs for all theorems. When coming across Zsigmondy's theorem, it seemed difficult to find a proof available on the internet.
I've been searching through MSE's old topics, but so far I only found proofs of the special case where $b=1$ (using the same notation as on the Wikipedia page). For instance in this thread on AoPS and this paper by M. Roitman (Theorem 3).
In Mathematical Excalibur, feb 2012, I found the reference [2] to another proof by Lola Thompon (namely http://www.math.dartmouth.edu/~thompson/Zsigmondy's Theorem.pdf), perhaps the same as on the AoPS site (?), but this web adress seems outdated. Searching through the main site didn't help much either...
By "an elementary proof" I mean whatever can be proved without abstract algebra, since it is meant to be understandable by students at secondary school without introducing high-school level mathematics. (I consider cyclotomic polynomials as elementary for example.) At least, I suppose there is an elementary proof that, however, might be quite long. (Correct me if I'm wrong and I'll cease my quest.)

REPLY [6 votes]: I have found the following published reference: Elementary proof of Zsigmondy's theorem by M. Teleuca, which also cleans up the proof of Birkhoff & Vandiver.
Artin also gives a proof in the general case (which seems to be discovered independently).