Primitive Prime Divisor

Zsigmondy's theorem

If $a>b⩾1$ are coprime integers, then for any integer $n⩾2$, there is a prime number $p$, called a primitive prime divisor, which divides $a^n-b^n$ and does not divide $a^k-b^k$ for any positive integer $k<n$, with the following exceptions:

• $n=2$, and $a+b$ is a power of $2$; or
• $(a,b,n)=(2,1,6)$.

\begin{proof} See here. \end{proof}

an Application

Pasted image 20240607215245.png