A modern reference to the Zsigmondy Theorem

M. Teleuca, Zsigmondy's theorem and its applications in contest problems, International Journal of Mathematical Education in Science and Technology Volume 44, 2013 - Issue 3, Pages 443-451, https://www.tandfonline.com/doi/abs/10.1080/0020739X.2012.714493?mobileUi=0&journalCode=tmes20

The abstract begins, "In this article, we present a detailed proof of Zsigmondy's theorem."

See also https://math.stackexchange.com/questions/660585/elementary-proof-of-zsigmondys-theorem and Moshe Roitman, On Zsigmondy primes, Proceedings of the American Mathematical Society, Volume 125, Number 7, July 1997, Pages 1913–1919, https://www.ams.org/journals/proc/1997-125-07/S0002-9939-97-03981-6/S0002-9939-97-03981-6.pdf

Indeed, it is very difficult to find Zsigmondy's theorem with a proof in a book. However, it is proved in Appendix B to Chapter 30 in

Berkovich, Ya. G.; Zhmudʹ, E. M. Characters of finite groups. Part 2. Translated from the Russian manuscript by P. Shumyatsky [P. V. Shumyatskiĭ], V. Zobina and Berkovich. Translations of Mathematical Monographs, 181. American Mathematical Society, Providence, RI, 1999.