Zsigmondy's theorem and its applications in contest problems

Abstract

In this article, we present a detailed proof of Zsigmondy's theorem. It proves the existence of primitive divisors of numbers of the form a ⁿ  − b ⁿ . And while this is an interesting result in itself, it is also a powerful tool and can be very useful at solving number theory problems that appear at various olympiads.

Keywords:

• number theory
• Zsigmondy's theorem
• cyclotomic polynomials
• olympiads