Zsigmondy's theorem

Question

Please give reference for the list of problems which can be solved using zsigmondy's theorem,

I solved some problems but I want some more problems to understand the theorem thoroughly.

Thanks in advance.

Answer 1

Here are some problems I found on the internet which could be solved by Zsigmondy's Theorem.

$\textbf{Problem 1}$ (Romania TST 1994): Prove that the sequence $a_n = 3^n − 2^n$ contains no three numbers in geometric progression.

$\textbf{Problem 2}$ (Italy TST 2003): Find all triples of positive integers $(a, b, p)$ such that $2^a + p^b = 19^a$.

$\textbf{Problem 3}$ (Folklore): Find all nonnegative integers $m, n$ such that $3^m − 5^n$ is a perfect square.

$\textbf{Problem 4}$ (BMO 2009 Problem 1): Find all $x, y, z \in \mathbb{N^3}$ such that: $5^x − 3^y = z^2$.

$\textbf{Problem 5}$ (Polish Mathematical Olympiad): Let $2 < p < q$ be two odd prime numbers. Prove that $2^{pq} − 1$ has at least three distinct prime divisors.

$\textbf{Problem 6}$ (Czech Slovakia 1996): Find all positive integers $x, y$ such that $p^x − y^p = 1$ where $p$ is a prime.

$\textbf{Problem 7}$ (Folklore): Find all positive integers $a, n > 1$ and $k$ for which $3^k − 1 = a^n$.

$\textbf{Problem 8}$ (IMO Shortlist 2002): Let $p_1, p_2, . . . , p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2···p_n} + 1$ has at least $4^n$ divisors.

Problem 7 and 8 are from: https://pommetatin.be/files/zsigmondy_en.pdf while rest of these problems are from: https://course-notes.org/sites/www.course-notes.org/files/uploads/math/zsigmondy_theorem.pdf, which is consists of many more problems. Both of these two articles are excellent even for preparing a strong theoretical background to the theorem itself.